Sunday, January 14, 2007

Spherical Cows

As you wander along a dark (aren't they all?) hallway, you come upon something that's a surprise even for this museum: normal light! A large portion of one wall of the hallway has been replaced by a large sheet of glass, which allows some of the light to leak out. Beyond the glass are three spheres which appear to be painted to look like cows in a field of grass that looks to be larger than the entire museum.

At least, that was what you thought until one of them somehow started rolling on its own, causing itself to spin around until a face came into view. It appeared to be smiling, and let out a friendly "Mooo!" You remember seeing something similar back at the information booth in the lobby and had assumed it to be simple decoration. Could that have been alive as well?

You give the cow a quick wave, and then move over to a panel on the wall, looking for an explanation of what's up with these cows. The panel explains it:

The Spherical Cow is what's known as a simplifying assumption. These are tools used by scientists to make fairly accurate predictions about world, without having to perform the extremely complex math inherent in, for instance, calculating the moment of inertia or drag coefficient of an oddly-shaped body such as the normal cow.

Such assumptions may seem like they lead to worse results, but this isn't entirely fair. The truth is that in many cases if they weren't used, we wouldn't be able to get any results. We also have to be sure to remember that our results aren't expected to mesh perfectly with reality, and the only way to know exactly what will happen in any given case is to test it out experimentally.

The problem is that a technique such as this is often used as an excuse for quacks to exercise shoddy science. Simplifying assumptions are used which completely undermine the entire endeavor. For instance, take the case of one particular Creationist argument from big numbers, as previously debunked by MarkCC. Look at the following passage:


(1) we will reckon the odds of evolving a new horse species from an earlier horse species.

(2) we assume only random copying errors as the source of Darwinian variation. Any other source of variation -- transposition, e.g., -- is non-random and therefore NON-DARWINIAN.

The first "Assumption" isn't so much an assumption as an outline of what they're doing here, but the second one is what I want you to look at. There are many types of mutations that can occur, but they're making the unfair simplifying assumption of only looking at random copying errors. I'll let MarkCC explain what's wrong with this:

Really, there are a lot of different sources of variation/mutation. At a minimum, there are point mutations, deletions (a section getting lost while copying), insertions (something getting inserted into a sequence during copying), transpositions (something getting moved), reversals (something get flipped so it appears in the reverse order), fusions (things that were separate getting merged - e.g., chromasomes in humans vs. in chimps), and fissions (things that were a single unit getting split).

In fact, this restriction a priori makes horse evolution impossible; because the modern species of horses have different numbers of chromasomes. Since the only change he allows is point-mutation, there is no way that his strawman Darwinism can do the job. Which, of course, is the point: he wants to make it impossible.

This assumption doesn't make it simpler to find results, it makes it impossible to find results, and they use this as the base of their argument that evolution therefore must be impossible. Of course, it wouldn't be impossible at all if they didn't make this assumption.

So, how does one go about determining if a simplifying assumption is reasonable? For this, I've created the following checklist:
  1. Is there a legitimate reason to make the assumption? That is, without it, would the involved math still be reasonable?
  2. Is the assumption used in some field other than strict mathematics (in which no assumptions are allowed at all in proofs)?
  3. Is the assumption a reasonable representation of reality?
  4. Does the assumption preserve all critical, relevant portions of the situation?
If the answers to all of these questions is "Yes," then it's most likely a reasonable assumption. Let's go over a couple of cases to show how this works. First, let's take the case of assuming a Spherical Cow when determining the gravitational attraction between it and Earth:
  1. If we didn't make the spherical assumption, we would be faced with an impossible integral of determining Earth's gravitational attraction to every atom of the cow's body. Answer: Yes
  2. This is a case of physics, so Yes.
  3. Well, cow's aren't normally particularly spherical, but the important consideration here is the distance involved. Compared to the distance to the center of Earth, the size of a cow is virtually a point, so we can treat it as a point mass. Also to note: spherical mass distributions will always behave just like point masses, so on the scale we're working with, the assumption gives us a good approximation of reality. Answer: Yes.
  4. Nothing important left out, so Yes.
Looks like this is a reasonable assumption. Now, let's take the Creationist assumption of only copying errors outlined above:
  1. Without this assumption, we'd have to consider what would happen with many different types of mutations. It does drastically complicate the problem, but it might not be impossible for a good mathematician. Of course, these people aren't good mathematicians, so I'll give them some leeway and answer Yes here.
  2. Evolutionary biology, so Yes.
  3. Definitely No. This is only one of a myriad of mutations, so excepting all of those divorces this from reality.
  4. All of the other mutation types are indeed important, but particularly fission. This is the one that makes the species transition possible at all, so it definitely qualifies as critical. A resounding No here.
The last two are big No's, so this assumption most definitely isn't reasonable.

In an interesting quirk of fate, it was recently found that spherical cows do indeed exist, though they're quite rare. Many of the Light variety have taken a liking to this museum's curator's alter ego, Infophile, while many of the Dark variety have of course similarly taken a liking to our curator himself. In their experiences with the cows, they've found out the following about how they get around their spherical shape, which would seem to be an impediment:

They're capable of rapidly shifting around their mass density within their bodies, which causes an imbalance, in turn causing them to roll. By rolling in small circles, they can also accomplish turning in any direction they wish. They're also capable of retracting a portion of their body and quickly jerking it back out, causing them to apparently jump. Some Spherical Cows have even been known to repeat this process upon landing, causing them to bounce even higher.

When eating and drinking, Spherical Cows must orient themselves so that their face is pointing directly downwards. Without their faces visible, this causes many people to mistake them for simply spheres painted cow-like.

The spherical shape of these cows does have one known survival advantage: They're immune to cow-tipping. Cow-rolling has been attempted by a few ambitious pranksters, but they've tended to underestimate the ability of Spherical Cows to turn on a dime and have thus become the victims of their own rolling.

* * * * *

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