Don't Forget the SB Challenge

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Just a reminder: the ScienceBloggers DonorsChoose challenge is not over yet. 10 GM/BM readers have already contributed over $1100. I can’t even begin to say how terrific I think that is. 9 of the proposals that I picked for the challenge have been fully funded!
Just to try to motivate other folks – here’s some of the things that I’d like to see get funded:

  • Pre-Calculus Text Books for some awesome students!. This is the biggest proposal that I included in the GM/BM challenge, and I don’t believe that we’ll manage to fund it just from SB readers. But this is a school with a class of kids who want to learn pre-calc, but have no textbooks. The proposal is to get enough textbooks for the all of the three classrooms full of kids who want to take the course. There’s nothing more disgraceful than a classroom full of hard-working students who’s school can’t even get them books. I put a couple of hundred of my own dollars (which I’m not counting in the SB challenge) towards this; let’s see how much we can do to help these kids.
  • Future Mathematicians in Need of Tools. This is a proposal to get math games and flashcards to help fourth graders learn basic math skills. The whole proposal is only $219, and 20% of that is already funded. There are a bunch more similar proposals for this kind of supply. Go look at the challenge and see if any of them catch your interest.

To contribute to these, just click on over to DonorsChoose.. And if you do contribute, don’t forget that SB is running a drawing to give away goodies to the people who’ve contributed. Just send a copy of the confirmation message that you get from DonorsChoose to “sb.donorschoose.bonanza@gmail.com”. You can see the stuff that we’re giving away over on Janet’s blog.
There’s also lots of other great science proposals that could use money. Janet’s got the rundown.

Ask an SBer: What makes a good science teacher?

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It’s that time of the week again, and a new “Ask an SBer” question is out. The question is: “What makes a good science teacher?”
As usual, since I’m the only math blogger around here, I’m going to shift the subject of the question a bit, to “What makes a good math teacher?”. The answer is similar, but not quite the same.
In my experience, what makes for a good math teacher is a few things:

  1. The ability to teach. This should go without saying, but alas, it doesn’t. There are an appalling number of folks out there who are brilliant mathematicians and genuinely nice people who have all of the other skills I’m going to mention, but have absolutely no concept of just how to get in front of a group of people and teach in a reasonable coherent way.
  2. Enthusiasm. Most people have an unfortunate sense that math is miserable drudgery. Teaching something mathematical, one of the most important things you can do is to just be genuinely enthusiastic – to make it clear that you love what you’re talking about, and that it’s something fun and exciting.
  3. Balance. The power of math comes from the way that it breaks things that you’re studying into simpler abstractions. Abstraction is the key to the value of mathematics. But it’s very easy to get caught up in the abstraction, and forget why you’re doing it. Good math teaching is a subtle act of balance: you’re studying abstractions, but you need to keep the applications of those abstractions in sight in a way that lets your students understand why they should care.

    There are two teachers that come to mind when I’m talking about this, both in mathematical specialties of computer science.
    One is Eric Allender, a professor at Rutgers University, who taught my first course on the theory of computation. ToC is a field that can get incredibly difficult, and can often push abstractions so far away from reality that it’s hard to see what the point of it is. Eric had everything that I said above nailed down perfectly: he had the ability to stand in front of a classroom full of people and explain difficult concepts in a way that made them comprehensible; and he caught us up in his enthusiasm for the subject, so that we caught on to why these difficult abstract things were interesting; and he always kept things grounded in a way where it was clear to us why we should care about it.
    The other is Errol Lloyd, a professor at the University of Delaware, why I did my PhD, and a member of my dissertation committee. Errol is a professor who studies algorithms – not quite as abstractly mathematical as ToC, but a subject that many computer science students dread. I certainly wasn’t looking forward to it coming into the class: my undergrad experience in the topic was awful. (The main thing I remember about it was the professor who seemed to only own one shirt, which he never washed. It was a running joke among the students, because every time we saw him, the shirt was dirtier. Same stupid blue turtleneck, which was almost more grey than blue by the end of the semester.) In contrast to the dreadful undergrad experience, Errol’s class was one of my favorite classes ever. Errol has the most astonishing teaching method I’ve ever seen. He doesn’t directly tell you anything: he gets up in front of the class, and starts asking questions. But the questions guide you through the process of discovering the subject that he’s teaching. And as he does it, he’s excited and happy and very, very kinetic, bouncing around the classroom, peppering different students with his questions. So as a student, you’re involved, and you’re caught up in his enthusiasm. (For those who understand what I’m saying: imagine a professor who can lead you through the process of inventing LR parsing from scratch, without ever telling you how to do it – just asking the right questions to force you to work through the problems that led to the invention of the LR parsing algorithms.)

Category Theories: Some definitions to build on

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Sorry, but I actually jumped the gun a bit on Yoneda’s lemma.

As I’ve mentioned, one of the things that I don’t like about category theory is how definition-heavy it is. So I’ve been trying to minimize the number of definitions at any time, and show interesting results of using the techniques of category theory as soon as I can.

Well, there are some important definitions which I haven’t talked about yet. And for Yoneda’s lemma to make sense, you really need to see some more examples of how categories express structure. And to be able to show how category theory lets you talk about structure, we need to plough our way through a few more definitions.

Initial Objects

Suppose we have a category, C. An object o ∈ Obj(C) is called an initial object in C if/f (∀ b ∈ Obj(C)), (∃ 1 f : o → b ∈ Arrows(C)).

In english: an object o is initial in C if there’s exactly one arrow from o to each other object in C. We often write “0C” for the initial object of a category C, or just “0” if it’s obvious what category we’re talking about.

There’s a dual notion of a terminal object: an object is terminal if there’s a exactly one arrow from every object in the category to it. Terminals are written “1C” or just “1”.

Given two objects in a category, if they’re both initial, they must be isomorphic. It’s pretty easy to prove: here’s the sketch. Remember the definition of isomorphism in category theory. An isomorphism is an arrow f : a → b, where (∃ g : b → a) such that f &ormd; g = 1b and g º f = 1a. If an object is initial, then there’s an arrow from it to every other object. Including the other initial object. And there’s an arrow back, because the other one is initial. The iso-arrows between the two initials obviously compose to identities.

Categorical Products

The product of two morphisms is a generalization of the cartesian product of two sets. It’s important because products are one of the major ways of building complex structures using simple categories.

Given a category C, and two objects a,b ∈ Obj(C), the categorical product a × b consists of:

  1. An object p, often written a×b;
  2. two arrows pa and pb, where p ∈ Obj(C), pa : p → a, and pb : p → b.
  3. a “pairing” operation, which for every object c ∈ C, maps the pair of arrows f : c → a and g : c → b to an arrow Pairc(f,g) : c → a×b. Pairc(f,g) is often written <f,g>c>.

Pairc must have three properties.

  1. pa º Pairc(f,g) = f.
  2. pb º Pairc(f,g) = g.
  3. (∀ h : c → a×b) Pairc(pa º h, pb º h) = h.

The first two of those properties are the separation arrows, to get from the product to its components; and the third is the merging arrow, to get from the components to the product. We can say the same thing about the relationships in the product in an easier way using a commutative diagram:

catprod.jpg

Fact or Fiction? Incredible Gullubility of the Clueless

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Off topic, but can’t resist commenting, for reasons that will become clear.
Over at Sadly No, Brad discusses a review of Glenn Reynold’s latest book, which includes some babble about “Transhumanism”.
The very first comment?

I was watching a show on the History Channel on Star Trek’s influence on scientific research- there’s a guy in Britain, as I recall, who is trying to connect everyone to the Internet, and engage in cyborging, etc.
This is, actually, rather more widespread an ideal than one would think, particularly among both neo-libs and communistic types; theoretically, a non-intrusive connection would “benefit everyone” and allow complete connection without the need of a government to work together… theoretically.
I think it’s a pile of horseshit, naturally- and will happily proclaim my reactionaryness about keeping my bits and pieces together as God intended them (flesh, unless they fail first).

Now, who do you suppose the British scientist he’s talking about might be?
I’ll give you a hint. Look at this.
Yes indeed, it’s that great visionary of modern Britain, John Lumic, CEO of Cybus Industries.
A fictional character.
A fictional character from a goofy, but terrific TV show, which has put together some really silly tie-in websites.
Lumic is the villain in a Dr. Who storyline that was recently broadcast in the UK. He’s the inventor of that old Dr. Who staple enemy, the Cybermen. The upgrades to connect people to the internet and give them cybernetic limbs are the first steps, in the storyline, to turn people into Cybermen.
From the “Thoughts from our CEO” of the Cybus Industries website:

People often ask me: “John Lumic, you’re the richest man on the planet, you control the media, the arms trade, computing, medical research, the Cybusnet™, the telecommunications industry and the space programme. Some even say you control governments. What is there left to do?” To which I reply: “The hardest job of all.”
Every one of us, great or small, rich or poor, important or unimportant, me or you, will wither and die whether we like it or not. Mortality is the universal enemy, and until mortality itself is beaten and subjugated, until I discover the cure for death, then my work on this world is incomplete.
I am developing, as we speak, a series of cybernetic Upgrades™ for the human body. A skin of flesh and a heart of metal that does not age and does not die. These will be offered to the Great British public for free, whether you like it or not. They are both beautiful and compulsory. To those who ask me if I regret not having children, I say: “These Upgrades™ are my children.”

Yeah, there are actually people who take that website seriously.
Personally, I just can’t wait to actually see the damned storyline that it ties into. It’s been a long time since we got to see a new cyberman episode of Dr. Who. The very first Dr. Who episode that I ever saw was a Cybermen story, and it hooked me but good.

Notices of the AMS special issue on Kurt Godel

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Harald Hanche-Olsen, in the comments on my earlier post about the Principia Mathematica, has pointed out that this months issue of the Notices of the American Mathematical Society is a special issue in honor of the 100th anniversary of Kurt Gödels birth. The entire issue is available for free online
I haven’t read much of the journal yet; but Martin Davis’s article The Incompleteness Theorem is a really great overview of the theorem abnd the proof, how it works, and what it means.

New York: the Politest City in the World!

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Off topic, but as a proud New Yorker, I can’t resist. Over on Feministe, zuzu posted a link to an article about New Yorkers, and how when it comes to genuine helpfulness, NY is the best city in the world..
Basically, Readers Digest did a series of experiments, where they actually observed people in different cities in the world. NYers were rude, but far and away the most helpful city-dwellers.
I particularly love one thing zuzu included, because it perfectly captures the spirit of NY to me. A friend of her described New Yorkers like so:

f you fall down on the sidewalk, they’ll help you up. They’ll laugh at you, and tell you you’re a fool, but they’ll help you up.

That’s my city all right.

Conservative Morons

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I know this is outside my usual subject area, but I just saw a spectacularly stupid quote from moronic conservative, and I just had to mock it.
From AllThingsConservative, an explanation of why we should thank god that Ronald Reagan was president:

If not for Ronald Reagan, and his vision and leadership, we would now be at the mercy of that lunatic in North Korea. Instead, we have a workable missile defense against the threat. Not only are we still working on a space-based system, but the technology now being deployed was developed due to the efforts started by Reagan.
We should definitely shoot down the test missile.

Y’see, if if weren’t for the fact that Ronald Reagan was president, we wouldn’t have spent billions of dollars building a missile defense system that doesn’t work. And if we didn’t have a totally non-functional missile defense system, then we wouldn’t have the kind of safety that only a non-functional missile defense system can provide.
After all, remember: this is the missile defense system that hasn’t been able to knock down a missile even under the most insanely favorable conditions. A system that has failed a majority of its tests, even when they knew exactly what trajectory the incoming target was going to take and the target missile was carrying a homing beacon.
Yup, we can all sit back and feel secure in the knowledge that we’re being protected by the genius of Ronald Reagan. Because if the North Koreans tell us exactly when they’re going to launch a missile at is; exactly what trajectory they’re going to use, and they’re kind enough to put a homing beacon on the warhead, we’ll have a slightly less than 50% chance of successfully shooting it down.
I feel so greatful, don’t you?

Homeopathy and Nosodes

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I’ve been meaning to write something about homeopathy at some point, because it’s just so wretchedly stupid. But until now, I haven’t sat down to actually do it, because it can seem rather like beating a dead horse: it’s just so over-the-top goofy, and the goofiness of it is so well documented that I wasn’t really sure what I had to add.
Then I came across something that was new to me.
As I’ve mentioned before, I’m a New Yorker. I live just north of the city in one of the Westchester suburbs. The anthrax attacks that happened a few years ago were a very big deal in my area – in particular, because one of my neighbors is a NYT reporter who at a desk in the room where one of the alleged anthrax letters was opened. (It turned out to be one of the faked copycat ones – an envelope full of talcum power, if I remember correctly.)
Anyway… I’ve been sick with a miserable sinus infection this week, and got a prescription for an antibiotic. The pharmacy that I use is, alas, rather heavy of the woo: they’ve got a homeopath and a naturopath providing consultations on the premises. But after trying rather a lot of different local pharmacies, I’ve found that it’s the only one where I haven’t been robbed or abused by the phramacist. (By robbed, I mean having them fill prescriptions with less pills than I’m paying for; since I use some expensive stomach medications, I’ve had $300 prescriptions with fully a tenth of what I was supposed to be given left out of the bottle. I’ll take a woo-ish pharmacist who’s honest with me, leaves me alone when I say I don’t want to hear the woo, and gives me what I pay for over a lying, cheating scumbag.)
Anyway… Heading down to my pleasant but rather woo-ish pharmacy, there’s a note on a bulletin board about nosodes, and how we should all be preparing safe quantities of nosodes for anthrax and smallpox attacks in order to protect our families? I’d never heard the term before. So when I got home with my non-woo medication, I hit the net to figure out what this stuff was.
According to The National Center for Homeopathy, nosodes are

homeopathic attenuations of: pathological organs or tissues; causative agents such as bacteria, fungi, ova, parasites, virus particles, and yeast; disease products; excretions or secretions

Translated: take either a sample of an infectious agent, or an infected tissue sample. Dilute it down to silly proportions using a magic shaking ritual (homeopathic attenuation, aka succussion), voila! You have a nosode.
What’s going on here? And why is it bad math?
The homeopathic shaking ritual is, basically, take something like a nosode. Mix it with water, in a 100 to 1 proportion, using the special magic shake. Now, take a sample of that mixture – and mix it with water again – 100 parts water to 1 part solution, and shake. Repeat many, many times. Many homepathic remedies use a 100 to 1 dilution repeated 20 times – the so-called “20C” dilution. The more times you repeat the magic shaking ritual, the stronger the alleged medicine becomes.
Normal homeopathic remedies are based on the dilution of substances that produce the same symptoms as the illness that they’re allegedly treating. So they’ll take some substance – a salt, an herb extract – and dilute it this way. So in a 20C dilution, you’re talking about 1 part active ingredient in 100^20 parts water – so you’re talking about one part active incredient in 1×10^400 parts water. This is also known as “pure water”. In a regular dose – several teaspoonsful of the diluted solution – you are almost certainly not getting a single molecule of the active ingredient in the dose.
This is the first piece of really bad math in it. The inventor of homeopathy had no idea about how many molecules were in water; it’s not even clear that he really knew what molecules were. (Homeopathy was invented in the 1820; the molecular theory of matter as we understand it was proposed in 1812.) But we do now know about that – and so we know, by a combination of simple arithmetic and the number of molecules involved, that it’s an undeniable fact that these dilutions are entirely eliminating the supposedly active ingredient. To insist on a magical effect from something which has been eliminated from the solution is just stupid.
The idea that somehow diluting a solution to the point where there’s probably not a single molecule of the active ingredient left creates a good medicine is stupid – and that diluting it more after there’s not a single molecule left is even stupider.
The claim of modern homeopathy is that there are basically magical propeties of water: that the magic shaking ritual leaves crystalline structures in the water that are based on the active ingredient, and that those magic structures somehow are the cause of the effectiveness of homeopathy. This belief is predicated on the notion that the basic particles of the homeopathic ingredient is small enough that it can leave a specific shape in the arrangement of molecules in water.
So – what about nosodes? Well, the argument for a nosode is that it’s basically a kind of a vaccine: you’re putting the specific infection agent into solution – either directly if you can identify and extract the agent; or indirectly by using an infected tissue sample if you can’t. Then you’re doing the dilution.
In particular, they claim that they can “vaccinate” you against anthrax using a nosode solution of anthrax. But anthrax isn’t a molecule. It’s a bacteria – and a largish one at that. The so-called “crystalline structures” of water that homeopaths propose as an active principle are at an entirely different scale: this is another error of arithmetic; something orders of magnitude larger than a water molecule is not going to interact with a water molecule in the same way as something of its own size: it’s like saying that if you dip your finger into a pile of dust, take it out, and then shake the dust around, that the dust can retain the shape of your finger. The size of the structure that you claim to form is so much larger than the things it’s formed from – it’s a very silly idea.
What’s particularly scary about this: these people are applying for homeland security funds to stockpile nosodes. They claim that they can provide “homeopathic vaccines” using nosodes for less money, and in less time than it would take to prepare real vaccines or treatments. And in the current political climate, they may very well be taken seriously, and get money that could have been used to buy or produce real remedies, rather than magic water.

Category Theory: Natural Transformations and Structure

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The thing that I think is most interesting about category theory is that what it’s really fundamentally about is structure. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate structures in a simple visual way. Morphisms express the structure of a category; functors are higher level morphisms that express the structure of relationships between categories.

In my last category theory post, one of the things I mentioned was how category theory lets you explain the idea of symmetry and group actions – which are a kind of structural immunity to transformation, and a definition of transformation – in a much simpler way than it could be talked about without categories.

It turns out that symmetry transformations are just the tip of the iceberg of the kinds of structural things we can talk about using categories. In fact, as I alluded to in my last post, if we create a category of categories, we end up with functors as arrows between categories.

What happens if we take the same kind of thing that we did to get group actions, and we pull out a level, so that instead of looking at the category of categories, focusing on arrows from the specific category of a group to the category of sets, we do it with arrows between members of the category of functors?

We get the general concept of a natural transformation. A natural transformation is a morphism from functor to functor, which preserves the full structure of morphism composition within the categories mapped by the functors.

Suppose we have two categories, C and D. And suppose we also have two functors, F, G : C → D. A natural transformation from F to G, which we’ll call η maps every object x in C to an arrow ηx : F(x) → G(x). ηx has the property that for every arrow a : x → y in C, ηy º F(a) = G(a) º ηx. If this is true, we call ηx the component of η for (or at) x.

That paragraph is a bit of a whopper to interpret. Fortunately, we can draw a diagram to help illustrate what that means. The following diagram commutes if η has the property described in that paragraph.

natural-transform.jpg

I think this is one of the places where the diagrams really help. We’re talking about a relatively straightforward property here, but it’s very confusing to write about in equational form. But given the commutative diagram, you can see that it’s not so hard: the path ηy º F(a) and the path G(a) º η<sub compose to the same thing: that is, the transformation η hasn’t changed the structure expressed by the morphisms.

And that’s precisely the point of the natural transformation: it’s a way of showing the relationships between different descriptions of structures – just the next step up the ladder. The basic morphisms of a category express the structure of the category; functors express the structure of relationships between categories; and natural transformations express the structure of relationships between relationships.

Coming soon: a few examples of natural transformation, and then… Yoneda’s lemma. Yoneda’s lemma takes the idea we mentioned before of a group being representable by a category with one object, and generalizes all the way up from the level of a single category to the level of natural transformations.

The Stupidity of Numerology, illustrated by Infinite Sequences

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I was glancing at the comments on the post that I linked to about “0.999…=1”. And one of them was such a wonderful example of crap numerology, which I enjoy laughing at, that I just had to repost it here:

VERY GOOD.
But there’s a couple tricks
you missed.
First, simple pattern
completion
1/9 = .11111—
2/9 = .22222—
3/9 = .33333—
4/9 = .44444—
5/9 = .55555—
6/9 = .66666—
7/9 = .77777—
8/9 = .88888—
and therefore by logical
extension
9/9 = .99999—
but of course, 9/9 = 1.
And then there are the
SPIRITUAL implications
.9 a soul
+ .09
+ .009 adding experience
+ .0009
+ .00009
!
! infinitely increasing
!
or the infinitely
repeating process
of growing greater
i.e. life