Wednesday, August 25, 2010

Futurama's Math Jokes

Those that love math will probably get a kick out of Futurama and their blink and you miss it math jokes.

Aleph-Null Plex as a theater name instead of a "multiplex." Now that's good. For those that don't know, Aleph-Null is a part of Set Theory, a mathematical concept described in the 1870s by Georg Cantor, a concept that is taught even at simple levels by the use of Venn diagrams (remember those?). According to him there are various types of infinities, and because Cantor was Jewish, he described by the Hebrew letter Aleph.

Aleph-Null (or Aleph-Zero) is used to describe the set with the smallest cardinality (or size of the elements in a set). It measures an infinity according to natural, ordinary counting numbers (excluding zero, negative numbers, and irrational numbers). Cantor made a distinction between transfinite and absolute infinity, in the sense that transfinite numbers are sets bigger than any finite set, yet they fall far short of absolute infinity. In fact, it's been demonstrated at least in classical cardinal mathematics that the sum of all ordinal numbers can't possibly exist, something called the Burali-Forti Paradox.

Think of the Burali-Forti Paradox like this. Take something that is meant to represent the sum of all ordinal numbers. Cantor was partial to the Omega symbol for religious reasons. Now, the concept you just created has all the properties of a number that can be listed in a set! There are some interesting ways to resolve this paradox, notably through use of different principles of set theory.

Friday, August 20, 2010

The cranks were right!

Well, it doesn’t happen that often, but there are occasions where crackpot science is right on the money...or perhaps a better way of putting it is there are occasions here and there that crackpot science coincides with what real science discovers.

I find it amusing when this happens. It’s like when paranoia turns out to be correct. I’m pretty young, but most of these went from crackpot pipe dream to real substantiated fact within my lifetime.

The King Cheetah – a Cryptozoologist actually caught something real

It’s like if Charlie Brown finally kicked that football. Good work, guys!

The first time I ever heard of the King Cheetah was on A&E reruns of “Arthur C. Clarke’s Mysterious World,” a documentary series hosted by Arthur C. Clarke, who took time off to perform series duty from his primary career as an author of autistic hard-SF novels, and his secondary career as a loathsome, lecherous pederast. It was on a show about weird and mysterious animals that have been sighted all over the world, and what made it special is not just the usual emphasis on giant ape-men and tourist-magnet lake monsters, but on weirder, less famous and more interesting creatures like the King Cheetah.

A cheetah with stripes along the spine and a giant black mane have been spotted in Africa since 1925, and one was even caught on a grainy home movie in the 1970s. Were they a figment of the imagination or an undiscovered species?

Eventually the conflict was resolved when a pair of Cryptozoologists, Lena and Paul Bottriell managed to photograph one in the 1970s.

In 1981, a cheetah in captivity in South Africa gave live birth to a pair of King Cheetahs. This resolved the species status of the King Cheetah: they weren’t a separate species, but some unusual mutation within the Cheetah family. Since then, a handful of King Cheetahs have been found (tragically, most were in the form of pelts).

Now, be honest: would you have expected something you first heard about on an Unsolved Mystery show turn out to be a star zoo show attraction? What next? Unicycle-riding Bigfoots?

The Vikings Were in America

Theories about how (insert culture here) reached America before Columbus are a dime a dozen and worth that, too…wispy and insubstantiated. The first time I ever read about Vikings in America was in an unsolved mystery casebook, and the cases right next to it wondered if there might be some truth to the Curse of Tutankhamun and if the Tunguska blast was from a crashed UFO. Considering the context, the idea that the Vikings might have been in America didn’t inspire confidence.

Then at some point in my middle school or high school years, it went from a crank theory to something “everyone knows,” that’s in every single textbook. New waves of artifacts at L’Anse aux Meadows in Newfoundland suggested to even the most hardened cynic that the Vikings reached America, including sod-buildings identical to those in Denmark and remains of shops for blacksmithing iron, something that was never practiced by Native Americans. And even more, scholarship for the Norse sagas improved and are being actively reread and taken seriously as historical documents instead of just as literature.

Zombies do exist!

I’m getting advising for it now, and man, I sure wish Wes Craven would turn my Masters Thesis into an awesome horror movie the way he did for Wade Davis when Wes made “The Serpent and the Rainbow,” one of the most terrifying films ever made.

Anthropologist and ethnobotanist Wade Davis discovered there was a truth to all the stories of the Zombie in Haiti: Houngan witch doctors in Haiti created a powder where the primary ingredient is from a fugu fish, a muscle relaxant that makes a person fall into a deathlike coma state. While in this state where they felt no pain, body functions were slowed and they could be mislabeled as dead by non-trained observers, the victim was buried alive. Later on the person exposed to the poison is dug up, and the witch doctor convinces them they died and were brought back from the dead. With other pharmaceuticals, witch doctors in Haiti often kept people as their “zombie” slaves for years and years, a combination of trance drugs, suggestion and cultural beliefs that created true mind control.

The fundamental conservatism of anti-science cranks (or, how we know for a fact the earth isn’t hollow)

Never underestimate the power of projection!

The most insufferable and sanctimonious thing about crackpot and fringe theorists is the insistence that they are new, daring, revolutionary, and rebellious, opposing the stodgy authority of science…when in reality it's pretty startling how downright traditionalist and conservative fringe theories are, and how little they change.

Two of the major fringe theories about the structure of the earth are very, very old theories that have since been superseded...yet cranks earnestly strive to keep them in circulation. In fact, I’d be hard-pressed to find a crank science theory that isn’t some old, recycled idea from science’s past.

Example #1: The Aquatic Ape Hypothesis

Cranks aren’t experts. They don’t keep up with the literature, with changing scientific ideas and trends. Check out this TED talk by this extra-cute old lady about the Aquatic Ape Theory. The basics of the Aquatic Ape theory is that the reason human ancestors are so different from the other apes is that we went through a stage that involved life extremely close to water.

…did you get all that? It’s a trippy theory, but it’s not scientific, because how can it be tested? Falsafiability is the single most important attribute in determining whether something is scientific or not. Every major theory in science has a way in which it can be proven untrue. Even something as central and bedrock to the biological sciences as the theory of evolution: as Neil “Tiktaalik” Schubin once put it, the one thing that can throw evolution out the window is finding “bunny rabbits running around in the Precambrian.”

But the single most revealing comment in her entire analysis is that scientists are still treating the Savannah Model of human evolution as if it’s still gospel. I’m surprised no one called her out on that. The Savannah Model was taken down as we learned more about Oligocene paleoclimate...way back in the 1960s.

The point is, since the 1960s, paleoanthropologists have come up with totally different models for human evolution that don’t involve the drying out of the Savannah at all. But the Aquatic Ape hypothesis exists because of flaws in the Savannah model, which science doesn’t use at all anymore. In other words, the field had moved on since the 1960s but the fringe theory hadn’t.

Example #2: The Hollow Earth

The absolute apotheosis of how traditionalist and resistant to change crank theories are, has to be the Hollow Earth . This is exactly what I mean when I say how clingy and unsinkable crank theories are, since the Hollow Earth model was more or less abandoned by real science as early as the 18th Century. At one point, it was a real scientific theory, created by none other than Sir Edmund Halley. Yes, that Sir Edmund Halley. He was so proud of his Hollow Earth theory that when it came time to take his portrait at age 80, the diagram in his hand was that of his theory of planetary concentric circles.

In brief, the Hollow Earth theory is one that claims the earth is an empty shell, with living things on the inside. There have been tons of great novels dealing with fanciful subterranean worlds (which usually have dinosaurs and lost civilizations), which makes this one of my favorites. This theory is sweetly endearing just because of how fanciful and over the top it is; in order to make sense of it one has to pretend the 20th Century didn’t happen. Just because it’s cool doesn’t mean it’s true, though.

With theories of this type, it is true that science and science education take some of the blame. (I said SOME of the blame, because the responsibility and onus for education falls on the individual.) Science isn’t some kind of magic, and while kids are shown the sliced apple core diagram of the earth from a very early age, it’s not explained how it is that we actually know the structure of the earth. Science, like math, is a process, and it’s more important to “show your work.”

Hollow Earth supporters are quick to point out that we haven’t gotten very far into the earth at all, a mere 7.6 miles. This does not however, mean that the structure of the deep earth is a mystery at all. Since lots of people show up to this place for professional wrestling style debunking of crackpots, here’s how we know that the earth is solid:

Seismic Evidence. Earthquakes are essentially kinetic energy, and like all forms of kinetic energy, they travel in waves through the earth. There are two major types of seismic waves: P-waves and S-waves. P-waves, or primary waves, are produced by the alternation between expansion and compression and travel the fastest from an earthquake. P-waves travel out the fastest, can move through solid, liquid and gas, and their shape is changed as they move through heterogeneous substances. By analyzing the velocity of primary waves, it’s possible to learn about the density and composition of substances by the reflections and refractions of p-waves within the earth, without having to drill miles and miles into the earth at all. Sweet, eh?

Now, take S-waves, or secondary or shear waves, another, more slower-moving type of wave transmitted only through solids. If you’ve ever grabbed a rope and yanked it, you’ve seen how an s-wave propagates. Obviously, these only move through matter in the solid state; through liquids and gases, they attenuate, which means they lose intensity very quickly.

Check out the visual aid to the right. It is possible to receive primary waves from an earthquake. If the earth was a shell instead of a sphere, transverse p-waves would not move in direct polarity from the point of origin THROUGH the earth, would it? Likewise, the fact there is a shadow zone where no secondary s-waves can be received was the first clue that there was a level of the inner earth that was entirely molten and liquid.

The observable operation of gravity and centrifugal motion. Why are objects of sufficient mass in the universe round, anyway? Planets and stars and so on. Carl Sagan put it best when he explained that roundness is a property of gravity distributing matter evenly, because it pulls matter with more or less equal strength in one direction; because of this, planets and stars are more or less spherical. In fact, the more extreme heights are only possible on smaller, lighter worlds with lower gravity. Mars has an extinct volcano, Olympus Mons, the size of the state of Missouri that is five times taller than Mt. Everest. On earth, gravity would have flattened a mountain like that out and spread it evenly.

What’s more, gravity pulls things toward a center of mass. There’s no way a person could walk on the “inside” of a shell. Isaac Newton proved this way back when, when he demonstrated that a shell can’t exert force pushing something to the inner mass of a shell. If you love math and physics, check out this explanation of the Newtonian shell theorem.

As any high school geometry student can tell you, a sphere is the shape with the highest ratio of surface area to volume. It’s for this reason that hot air balloons are round, for instance. Compression due to internal gravity acting evenly would by necessity turn a world spherical because of hydrostatic equilibrium. A hollow-shell earth would simply collapse.

And then there’s the property of centrifugal motion. Earth’s outer crust is composed principally of lighter elements, like silicon and aluminum, whereas the heavier elements like iron and nickel are mainly drawn to the center, as in a centrifuge, swirling as in the early earth, while the mantle remains silicate-poor. This real world observable phenomenon is consistent with p-wave readings. Why would gravity turn the earth into a shell when it doesn’t work that way?

The earth’s mass is consistent with it being solid-through and not a shell. Try to imagine this: the force of gravity is a constant 9.8 m/s, and the force of acceleration due to gravity is 9.8 m/s^2. The entire mass of the earth is required to pull down a ping-pong ball, and everything on earth falls at the same rate. If we know the force of gravity something exerts it’s possible to calculate the mass. Visit this link for an explanation on how the mass of the earth is calculated. In fact, visit it anyway, as it is one of the more extraordinary calculations in the history of science.

Example #3: The Expanding Earth

The landmasses of the earth fit perfectly together? Gee whiz, thanks for telling us, Neal Adams! Oh, if only there was theory in geology that would explain why that is…!

Did I say that the Hollow Earth was my favorite crank theory? I change my mind, this one is, because its most visible defender is a comic book artist best known for his work on Batman, Neal Adams. Visiting Neal Adams’s website is great because the “amateur geology” stuff is placed side by side with great Batman concept art.

Like the Hollow Earth, this is another theory that was originally scientific but science discarded its usefulness and utility when it just moved past and beyond it. Once again, it is the fringe theorists that are the ones that are stodgy and resistant to change. The idea that the earth is expanding was first proposed way back in the 1950s by the Australian geologist S. Warren Carey as a pre-plate tectonics way of explaining some of the overwhelming evidence that would later be shown as proof of plate tectonics. S. Warren Carey, despite his stubborn insistence and clinginess to his pet theory, was not a crank: he was a real geologist and a real scientist at the University of Tasmania, a member of the last generation that could produce opposition to plate tectonics with any intellectual honesty, but science eventually moved past him and all the proofs he provided for his expanding earth eventually made the acceptance of continental drift over the static model that much easier.

Watch this video by Neal Adams where he makes the case for the expanding earth:

My favorite is the part where he explains that India just broke off and crashed into Asia but there was no explanation why. Neil even makes fun of this idea by asking if there were rocket motors placed on the end of the Indian subcontinent that jetted it up there. This is the best evidence yet for this theory as another conservative holdover from pre-plate tectonics days. One of the reasons it took so astonishingly long for the consensus in Geoscience to turn toward continental drift was that there was no sign of the mechanism by which drift took place. However, eventually the engine for it was discovered: seafloor spreading from the mid-oceanic ridge system. Saying that there’s no explanation for why India moved is a woefully out of date statement out of step with current geology. In fact, the Mid-Oceanic Ridge System that pushed India away is still there off the coast of Africa!

Neil goes in detail from his website, here:

“India, Geology says, tore itself off of Africa, rode up the Indian Ocean and crashed into Asia, against all laws, and Geologists show us this in books and on TV everywhere.

Tore itself off!! Just imagine. Took a trip like a plow goes from one side of a field to the other plowing up the Earth, and yet if you look at the oceanic plate there is no evidence of this happening. None! Then it crashed into Asia. Ah, crashed.”

Actually Neil, yeah it did, because there is evidence of deformation at collision boundaries for the Himalayas. By the way, there’s plenty of seafloor sediment that was upraised by the continental collision consistent with dating for the subcontinent’s movement.

The earth, is not in fact growing at all. No increase in the earth’s size has been reported through any geodesic technique. (Remember that?) On the other hand, the movement of continents have been measured by satellite observation. At no point in history has it ever been demonstrated that the earth is growing. What’s more, mass accretion produces heat, which has never been measured or found.

No other body in the universe is growing at a measurable rate. Have a look at this video from NASA Lunar Reconnaissance about how the Moon is actually shrinking.

Subduction is in fact, happening. Take the proof from satellites at points of subduction, which show often 60 miles or more of oceanic crust vanishing. Island chains form at (most) subduction points in arc-shapes, because of accretionary wedges where magma bubbles up only from subduction zones. Take the magmatic rock and date it if you like, and you’ll see that the source of magma that produces them is pretty consistent.

Example #4: Lemuria

Lemuria is an idea with a closely related history to the expanding earth, in that it began as a legitimate scientific hypothesis that ended when it was eventually superceded by continental drift…before it fell into the hands of crackpots.

One of the greatest proofs that living things were not created at a single point by a designer but instead are produced by modification and descent from other living things is that even in similar habitats, the living things there are more like the animals from the surrounding era. Take for instance, caves: there is no environment more similar than caverns. Two caves on other sides of the planet have more in common with each other than the environment outside. However, species of cave fish are more likely to be related to fish in streams near the caverns than the cave fish in more distant caverns.

When evolution first became accepted, it became necessary to explain similar fossils found in different regions of the world for the first time. Take lemurs, for instance. Though lemurs are only found alive in Madagascar, fossil lemurs were discovered in India and Africa. A provisional, tentative hypothesis was proposed in 1864 by Phillip L. Sclater, who said that a whole bunch of coincidences between wildlife between these areas could be explained by a long-lost land bridge that he called “Lemuria,” where presumably the lemurs crossed over. By the way, Philip Sclater was a great ornithologist and one of the first to advance a few models of ancient biogeography. In the end, the similarities between the ancient regions, including lemurs, were explained by continental drift.

That didn’t stop crackpot believers in Theosophy and other trendy 19th Century spiritualists from jumping to conclusions about Lemuria and making things up about it, though! All this from a barely-there bit of biogeography that was abandoned by real scientists as far back as the 1880s. Does anyone still think that crank science isn’t overly traditionalist?

Inventing a continent out of thin air to explain the absence of fossils is a time-tested tradition. To explain the (then) “abrupt” appearance of Cro-Magnon man, it was conjectured by crackpot supreme Lewis Spence in the 1920s that Cro-Magnon must have evolved on Atlantis. The Nazis liked this idea a little too much, and it’s possible to see it appear here and there on white power and racist websites.

The one guy inspired by the idea of Atlantis as the home of a stone age Cro-Magnon culture was Robert E. Howard, in my view the second-best Weird Tales writer after Clark Ashton Smith. He created his hero Kull the Conqueror as a stone age Atlantean!

Example #5: Michael Behe, the bacterial flagellum, and the evolution of blood clotting

It’s possible to write entire books to explain where Michael Behe’s “Darwin’s Black Box” went wrong…and some people have! Ken Miller, for instance. As the embarrassing and at times laughable “Darwin’s Black Box” is the closest thing to an actual scientific challenge that evolution has received in some time, it has been over-scrutinized out of a desire to set the record straight.

The book was written in 1993, and at the time, there had been absolutely no work done explaining the evolution of the blood clotting mechanism, which Behe explained was far too complex to naturally evolve and required a designer. Since that time, real, legitimate science has explained how something like that could have developed naturally. Check out Ken Miller’s website for an explanation. The point isn’t that this is how it happened though, the point is this is how it could have happened, which demolishes the entire idea of the necessity of a supernatural force.

And yet, despite the fact biology has changed, to date Michael Behe has never rewritten his 1993 book to take into account any new findings by science, including those spurred by his own criticisms of it. "Darwin's Black Box" is still published in more or less the same form now as it was in 1993 without retractions.

The next time someone talks about how stodgy and authoritarian and slow to change science is, that person should take a long look in the mirror. Science changes all the time. Crank science reuse the same ideas over and over.