Tuesday, March 31, 2009

Review of "The God Delusion"

I struggled a bit trying to rationalize posting this on my math-oriented blog. I finally came to the conclusion that (a) the book in question is largely biology science-themed, (b) it regards a subject that does in fact make me pretty angry, and (c) by the end of the review I do touch on topics of probability and statistics. Hence the posting here.

I've read several chapters of Dawkins' "The God Delusion" and I've got to say that it's disappointing. It's a worthwhile project ("consciousness raising" on why it's admirable to be an atheist), one that I've wanted to do myself in the past, but this doesn't quite fit the bill. Mostly it's a matter of style. It's simply to wordy; it's too discursive; it's too English. Dawkins seems unable to go more than a a single page without some lengthy outside quote; it feels like I just barely get into his train of thought before having to repeatedly jump into some other person's anecdote, poem, or metaphor. I used to take pleasure in nonstop tangents and wordplay like this, but I've found that my patience for it has died out.

I need something that's a bit more punchy, personal, and directly to the point. I would prefer a manifesto and we don't get that here. Dawkins clearly demonstrates a great deal of literary and cultural knowledge, but I find it altogether distracting. In addition, the foils that he's primarily skewering generally seem to be a batch of kindly, woolly-headed, liberal English archbishops, which seem like very faint opposition. Apparently one of the most common clerical responses that Dawkins hears is "Well, obviously no one actually believe in a white-bearded old man living in the sky anymore", which seems entirely off-topic to someone such as myself who lives in American society. He feels compelled to say things like "This quote is by Ann Coulter, who my American colleagues assure me is not a fictional character from the Onion," which again, is completely distracting and quizzical to the American reader.

I'll say this: Dawkins has great book titles. "The God Delusion" sounds like exactly something I'd been looking for, perhaps an explanation or theory of exactly why so many people's brains cling to religion. But frankly that's not what you find between the covers. The keystone Chapter 4 is titled, "Why There Almost Certainly is No God". That sounds compelling, and I could almost start sketching the chapter out in my head, using the modern statistical science of hypothesis testing as a model. But unfortunately the entirety of the chapter is taken up by Dawkins cheerleading for why the theory of natural selection is so great. Great it certainly is. But at best this chapter explains why God is unnecessary for the specific purpose of explaining the evolution of species. People use the idea of God for many, many purposes beyond that, and I think that a far more offense-directed argument needs to be made to fulfill the promise of Chapter 4.

Given Dawkins' focus on biological science and evolution, he has a razor-sharp sensitivity to arguments that "Such-and-such an organ is so complicated, it must have been designed by God"; he spends swaths of several chapters fighting them. Okay, that's a reasonable thing to be irritated by, but here's two observations. One is that I can summarize his argument in a single line. The response to any cleric's "What is the probability that organ X or universe Y could have appeared spontaneously?" should always be "Enormously greater than the probability that a sentient, all-knowing, omnipotent, thought-reading, personally attentive, prayer-answering God could have appeared by chance!" There, I just saved you about 3 chapters.

Secondly, I cannot help but take away the impression that we're fundamentally winning against such arguments. Dawkins makes a good point that a "mystery" to a scientist represents the starting point for an intriguing research project; whereas for a religious person it is a stopping point whose dominion must be reserved for God (historically, complete with threats of violence against exploration). But clearly the "God of the gaps" proponents are being pushed further and further back, perhaps even with greater velocity over time. Whereas previously they would point to organs such as an eye or wing as being impossible to evolve (and since having had the opposite be demonstrated), they have now, according to Dawkins examples, retreated to areas such as microbiology and the flagellum of bacteria. Presumably next will be quantum physics, and beyond that, some unidentifiable regress. My point here is that Dawkins' examples seem to take the emergency out of the issue, and at least from his focus on biological science, it seems like there's little we need to do to disprove God except to support ongoing biology research. I suppose that's good news, but I was looking for more of a direct call-to-action.

In Chapter 5 ("The Roots of Religion") Dawkins has some speculation on the question of "Why does religion exist?". To me, I felt like this was very specifically the promise of a book title "The God Delusion". But Dawkins has no specific thesis, he only has a loose collection of a half-dozen tentative speculations. The most tantalizing are the sections called "Religion as a by-product of something else" and "Psychologically primed for religion" (the centerpiece being, maybe children are mentally wired to implicitly trust what their parents say, so as to pass on key survival skills, and that leaves our species vulnerable to mind-viruses such as religion). It's an intriguing section, but it's short, Dawkins doesn't develop it greatly, nor does he stake out a specific position for it. My preference would be for him to have developed a specific, detailed thesis on the subject before presenting it in a book called "The God Delusion".

In summary: A commendable project, a great title, but a disappointing and distracting read for the American reader.

Friday, March 20, 2009

The Oops-Leon Particle

I think this is a great 3-paragraph story:

http://en.wikipedia.org/wiki/Oops-Leon

In short, in 1976 Fermilab thought it discovered a new particle of matter, but turned out to be a mistake. It was originally called the "upsilon", but after the mistake was caught, it was referred to as the "Oops-Leon", in a pun on the lead researcher, Leon Lederman. I love that wordplay.

The other thing I love is that, like all modern science, the mistake is partly due to statistics, which we must understand as being based on probability. Looking at a spike in some data, it was calculated that there was only a 1-in-50 chance for it not to have been caused by a new particle (that is, a P-value). But with further experimentation it turned out that that was a losing bet; it actually had been some random coincidence that caused the data spike.

That's the kind of thing you need to accept when using inferential statistics; all the statements are fundamentally probabilistic, and some times you're going to lose on those bets (and hence so too with all modern science). Apparently the new standard before publishing new particle discoveries is now 5 standard deviations likelihood, or more 99.9999% likelihood that your claim is correct.

And you know what? Someday that bet will also be wrong. Such is probability; so is statistics; and hence so is science.

Friday, March 6, 2009

PEMDAS: Terminate With Extreme Prejudice

Wednesday night, I walk into a lecture room for my first evening algebra class of the spring. And what do I see on the chalkboard? Some motherfucker has oh-so-carefully written out the PEMDAS acronym, with each associated word in a column sequence. In fact, that's the only thing he's got on the board after a presumably 2-hour lecture.

So, now it's time for my official AngryMath "Kill the Shit Out of PEMDAS" blog posting.

It's a funny thing, because I'd never heard of the PEMDAS acronym until I started teaching community college math. None of my friends had ever heard of it; artists, writers, engineers, what-have-you, from Maine or Massachusetts or Indiana or France or anywhere. But for some reason these urban schools teach it as a memory-assisted crutch for sort of getting the order of operations about halfway-right (PEMDAS: Parentheses, Exponents, Multiplying, Division, Addition, Subtraction.)

But the problem is, it's only half-right and the other half is just flat-out wrong. Wikipedia puts it like this ( http://en.wikipedia.org/wiki/Order_of_operations ):

In the United States, the acronym PEMDAS... is used as a mnemonic, sometimes expressed as the sentence 'Please Excuse My Dear Aunt Sally' or one of many other variations. Many such acronyms exist in other English speaking countries, where Parentheses may be called Brackets, and Exponentiation may be called Indices or Powers... However, all these mnemonics are misleading if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order addition first, subtraction afterward would give the wrong answer..."

In my experience, none of the students who learn PEMDAS are aware of the equal-precedence (ties) between the inverse operations of multiplication/division and addition/subtraction. Therefore, they will always get computations wrong when that is at issue. (Maybe prior instructors managed to scrupulously avoid exercises where that cropped up, but I'm not sure how exactly.)

Here's a proper order of operations table for an introductory algebra class. I've taken to repeatedly copying this onto the board almost every night because it's so important, and the PEMDAS has caused so much prior brain damage:

  1. Parentheses
  2. Exponents & Radicals
  3. Multiplication & Division
  4. Addition & Subtraction
Notice that each operation (after parentheses) is linked with its exact opposite (inverse) in equal precedence. Of course, you have to emphasize that (a) "parentheses" means "stuff inside the parentheses", and (b) in each stage you'll go left-to-right in the expression doing either the operation or its inverse in any order. In contrast, PEMDAS gets parentheses right (and that's pretty much it); but among its flaws are (1) leaving out radicals as the inverse of exponents, (2) overlooking that multiplication & division are tied, and (3) overlooking that addition & subtraction are tied.

An example I use in class: Simplify 24/3*2. Correct answer: 16 (24/3*2 = 8*2 = 16, left-to-right). Frequently-seen incorrect answer: 4 (24/3*2 = 24/6 = 4, following the faulty PEMDAS implication that multiplying is always done before division).

If you're looking at PEMDAS and not the properly-linked 4-stage order of operations, you miss out on all of the following skills:

(1) You solve an equation by applying inverse operations (i.e., cleaning up one side until you've isolated a variable). If you don't know what operation inverts (cancels) another, then you'll be out of luck, especially with regards to exponents and radicals. Otherwise known as "the re-balancing trick", or in Arabic, "al-jabr".

(2) Operations on powers all follow a downshift-one-operation shortcut. Examples: (x^2)^3 = x^6 (exp->mul), sqrt(x^6)=x^3 (rad->div), x^2*x^3 = x^5 (mul->add), x^5/x^3 = x^2 (div->sub), 3x^2 +5x^2 = 8x^2 (considering a shift below add/sub to be "no operation"). If you don't see that, then you've got to memorize what looks like an overwhelming tome of miscellaneous exponent rules. (And from experience: No one succeeds in doing so.)

(3) Distribution works with any operation applied to an operation one step below. Examples: (x^2*y^3)^2 = x^4*y^6 (exp across mul), (x^2/y^3)^2 = x^4/y^6 (exp across div), 3(x+y) = 3x+3y (mul across add), sqrt(x^2*y^6) = x*y^3 (rad across mul), etc. However, the following cannot be simplified by distribution and are common traps on tests: (x^3+y^3)^2 (exp across add), sqrt(x^6-y^6) (rad across sub), etc.

(4) All commutative operations are on the left, all non-commutative operations are on the right (the way I draw it). Also, any commutative operation applied to zero results in the identity of the operation immediately below it. Examples: x^0 = 1 (the multiplicative identity), x*0 = 0 (the additive identity), x+0 = x (no operation), etc. The first example is usually forgotten/done wrong by introductory algebra students.

(5) The fact that each inverse operation generates a new set of numbers (somewhat historically speaking). Examples: Start with basic counting (the whole numbers). (a) Subtraction generates negatives (the set of integers). (b) Division generates fractions (the set of rationals). (c) Radicals generate roots (part of the greater set of reals).

(6) Finally, per my good friend John S., perhaps the most important oversight of all is that PEMDAS misses the whole big idea of the order of operations: "More powerful operations are done before less powerful operations". I write that on the board, Day 1, even before I present the basic OOP table. It's not a bunch of random disassociated rules, it's one big idea with pretty obvious after-effects. (See John's MySpace blog.)

So as you can see, PEMDAS is like a plague o'er the land, a band of Vandals burning and pillaging students' cultivated abilities to compute, solve equations, simplify powers, and see connections between different operations and sets of numbers. If you see PEMDAS, consider it armed and dangerous. Shoot to kill.

Never-Ending Amazement

I started my spring semester's classes in the last few days, including two introductory algebra classes. It's possibly the best and most powerful start to a semester I've ever had; I got an extraordinarily good vibe from all my classes. So that's a good thing.

So here's a quick observation I've said aloud several times. For me, teaching college math is a continual exploration of the things people don't know. If I listen really carefully, I continually discover that the most basic, fundamental ideas imaginable are things that a lot of people in a community college simply never encountered. Whenever I start teaching a class and think "oh, christ, that's so basic it'll bore everyone to tears, skip over that quickly," I discover at some later point that a good portion of people have never heard of it in their life.

That's actually a good thing. It keeps me interested with this ongoing detective work I do to see exactly how far the unknowns go. And it provides the opportunity to share in the never-ending amazement, through the eyes of a student who never saw something really fundamental.

Here's one from this week (which I got to run in each of two introductory algebra classes). We're going to want to simplify expressions, with variables, even if we don't know what the variables are. ("Simplifying Algebra", I write on the board.) To do that, we can use a few tricks based on the overall global structure of numbers and their operations. I'm about to give 3 separate properties of numbers, here's the first. ("Commutative Property", I write on the board.)

Let's think about addition, say we take two numbers, like 4 and 5. 4 plus 5 is what? ("9", everyone says.) Now, if I reverse the order, and do 5 plus 4, I get what? ("9", everyone says.) Same number. Now, do you think that will work with any two numbers? With complete confidence, almost everyone in a room full of 25 people all say at once, "No, absolutely not."

So of course, I'm sort of thunder-struck by this response. Okay, I say, I gave you an example where it does work out, if you say "no" you need to give me an example where it doesn't work out. One student raises his hand and says "if one is positive and one negative". Okay I say, let's check 1 plus -8 ("-7"). Let's check -8 plus 1 ("-7"). So it does work out. Now do you think it works for any two numbers? At this point I get a split-vote, about half "yes" and half "no".

Okay, what else do you think it won't work for? One student raises her hand and says (and I bless her deeply for this), "if it's the same number". Uh, okay, let's check 5 plus 5 ("10"). And if I flip that around I again have 5 plus 5 ("10"). So now do you think it works for any two numbers?

At rather great length I finally get everyone agreeing "yes" to the Commutative Property of Addition. And it's obviously a point that no one in the class had ever realized before, that addition is perfectly symmetric for all types of numbers. You can sort of see a bit of a stunned look on some people's faces that they hadn't realized that before. Isn't that just incredibly amazing?

After that, we get to think about the Commutative Property of Multiplication (pointing out that commutativity does not work for subtraction or division), look at the Associative and Distributive Properties, do some simplifying exercises with association and distribution, and so on and so forth. But the thing I can't get over this week is that something so simple as flipping around an addition and automatically getting the same answer can, all by itself, be an enormous revelation if you listen closely enough.