This weeks SB question: What else would I do with my life?

As usual, once a week, the Seed folks send all of us a question from one of the SB readers:

Assuming that time and money were not obstacles, what area of scientific research, outside of your own discipline, would you most like to explore? Why?

I’ve actually got two answers to that question.

First up: theoretical physics. I’m fascinated by the work that’s trying to unify quantum mechanics and relativity: string theory, the shape of extended dimensions, etc. The problem is, I think that this answer is probably cheating, even though it’s my second choice after what I’m doing now. Because what attracted me to what I’m doing is the math: computer science is a science of applied math with a deep theoretical side; and what attracts me to physics is also the beautiful deep math. In fact, the particular parts of physics that most interest me are the parts that are closest to pure math – the shape of dimensions in string theory, the strange topologies that Lisa Randall has been suggesting, etc.

If that’s cheating, and I really have to get away from the math, then I’d have to say evolutionary development, aka evo-devo. Around holiday time last year, PZ posted a list of books for science geeks, and one was by a guy named Sean Carroll (alas, no relation) on evolutionary development. I grabbed the book on his recommendation – and the ways that gene expression drives the development of living things, the way you can recognize the relationships between species by watching how they form; the way you can use the relationships between species to explore how features evolved – it’s just unbelievably cool.

Earth as the center of the universe? Only if you use bad math.

One of my favorite places on the net to find really goofy bad math is Answers in Genesis. When I’m trying to avoid doing real work, I like to wander over there and look at the crazy stuff that people will actually take seriously in order to justify their religion.

In my latest swing by over there, I came across something which is a bizzare argument, but which is actually interesting mathematically. It’s an argument that the earth (or at least the milky way) must be at the center of the universe, because when we look at the redshifts of other objects in the universe, they appear to be quantized.

Here’s the short version of the argument, in their own words:

Over the last few decades, new evidence has surfaced that restores man to a central place in God’s universe. Astronomers have confirmed that numerical values of galaxy redshifts are ‘quantized’, tending to fall into distinct groups. According to Hubble’s law, redshifts are proportional to the distances of the galaxies from us. Then it would be the distances themselves that fall into groups. That would mean the galaxies tend to be grouped into (conceptual) spherical shells concentric around our home galaxy, the Milky Way. The shells turn out to be on the order of a million light years apart. The groups of redshifts would be distinct from each other only if our viewing location is less than a million light years from the centre. The odds for the Earth having such a unique position in the cosmos by accident are less than one in a trillion. Since big bang theorists presuppose the cosmos has naturalistic origins and cannot have a unique centre, they have sought other explanations, without notable success so far. Thus, redshift quantization is evidence (1) against the big bang theory, and (2) for a galactocentric cosmology, such as one by Robert Gentry or the one in my book, Starlight and Time.

This argument is actually an interesting combination of mathematical cluelessness and mathematical depth.

If you make the assumption that the universe is the inside of a giant sphere, expanding from a center point, then quantized redshift would be pretty surprising. Not just if you weren’t at the center of the universe – if the universe is an essentially flat shape, then a quantized redshift is very hard to explain. That’s because a quantized redshift in a “flat” universe would imply that things were expanding in a sequence of discrete shells, which would be quite a strange discovery.

But: if for some reason it was quantized, then no matter where you are, you will continue to see some degree of quantization in the motion of other objects. What you’d see is different redshifts – but they’d appear in a sort of stepped form: looking in any particular direction, you’d see a series of quantized shifts; looking in a different direction, you’d see a different series of shifts. The only place you’d get a perfectly uniform set of quantized shifts would be in the geometric center.

What the AiG guys ignore is the fact that in a flat geometry, the fact of quantized redshifts is incredibly hard to explain. Even if our galaxy were dead center in a uniform flat universe, the fact is, quantized redshifts – which imply discrete shells of matter in an expanding universe – are incredibly difficult to explain.

The flat geometry is a fundamental assumption of the AiG guys. If the universe is not flat, then the whole requirement for us to be at the center of things goes right out the window. For example, if our universe is the 3-dimensional surface of a four dimensional sphere – then if you see a quantized shift anywhere, you’ll see a quantized redshift everywhere. If fact, there are a lot of geometries that are much more likely to present a quantized redshift, and none of them except the flat one require any strange assumptions like “we’re in the dead center of the entire universe”. So do they make any argument to justify the assumption of a flat universe? No. Of course not. In fact, they just simply mock it:

They picture the galaxies like grains of dust all over the surface of the balloon. (No galaxies would be inside the balloon.) As the expansion proceeds, the rubber (representing the ‘fabric’ of space itself) stretches outward. This spreads the dust apart. From the viewpoint of each grain, the others move away from it, but no grain can claim to be the unique centre of the expansion. On the surface of the balloon, there is no centre. The true centre of the expansion would be in the air inside the balloon, which represents ‘hyperspace’, beyond the perception of creatures confined to the 3-D ‘surface’.

That’s the most substantive part of the section where they “address” the geometry of the universe. It’s not handled at all as an issue that needs to be seriously considered – but just as some ridiculously bizzare and impossible notion dreamed up by a bunch of eggheads looking for excuses to deny god. Even though it explains exactly what they’re trying to say can’t be explained.

But hey, let’s ignore that. Even if we do assume something ridiculous like a flat universe with us at the center, the quantized redshift is surprising. They specifically quote one of the discoverers of the quantization of redshift making this point; only they don’t understand what he’s saying:

‘The redshift has imprinted on it a pattern that appears to have its origin in microscopic quantum physics, yet it carries this imprint across cosmological boundaries.’ 39

Thus secular astronomers have avoided the simple explanation, most not even mentioning it as a possibility. Instead, they have grasped at a straw they would normally disdain, by invoking mysterious unknown physics. I suggest that they are avoiding the obvious because galactocentricity brings into question their deepest worldviews. This issue cuts right to the heart of the big bang theory–its naturalistic evolutionist presuppositions.

This is a really amazing miscomprehension here. They’re so desparate to discredit scientific explanation that they quote things that mean the dead opposite of what they say it means. They want it to say that the quantized redshift is unexplainable unless you believe that our galaxy is at the center of the universe. But that’s not what it says. What it says is: quantized shift is a surprising thing at all. It doesn’t matter where in the universe we are; if we’re in an absolute center (if there is such a thing), or if we’re on an edge, or if we’re in a random location in a geometry without an edge: it’s surprising.

But it is explainable by the very theories that they’re disdaining as they quote him; and in fact, his quote explains it. The best theory for the quantization of the redshift is that in the very earliest moments of the universe, quantum fluctations created a non-uniformity in the distribution of what became matter in the universe. As the universe expanded, that tiny quantum effect eventually ended up producing galaxies, galactic structures, and the quantized distribution of objects in the observable universe.

The quotation about the redshift pattern isn’t attempting to explain away some observation that suggests that we’re at the center of the universe. It’s trying to explain something far deeper than that. Whatever the shape of the universe, whatever our location in the universe, whether or not the phrase “the center of the universe” has any meaning at all, the quantized redshift is an amazing, surprising thing. And what’s particularly exciting about it is that this very large-scale phenomenon is a directly observable result of some of the smallest-scale phenomena in our universe.

Aside from that they engage in what I call obfuscatory mathematics. There are a bunch of equations scattered through the article. None of the equations are particularly enlightening; none of them actually add any real information to the article or strenghten their arguments in any way. They’re just there to add the gloss of credibility that you get from having equations in your article: “Oh, look, they must know what they’re talking about, they used math!”.

Some Basic Examples of Categories

For me, the frustrating thing about learning category theory was that
it seemed to be full of definitions, but that I couldn’t see why I should care.
What were these category things, and what could I really talk about using this
strange new mathematical language of categories?

To avoid that in my presentation, I’m going to show you a couple of examples up front of things we can talk about using the language of category theory: sets, partially ordered sets, and groups.

Sets as a Category

We can talk about sets using category theory. The objects in the category of sets are, obviously, sets. Arrows in the category of sets are total functions between sets.

Let’s see how these satisfy our definition of categories:

  • Given a function f from set A to set B, it’s represented by an arrow f : A → B.
  • º is function composition. It meets the properties of a categorical º:
    • Associativity: function composition over total functions is associative; we know that from set theory.
    • Identity: for any set S, 1S is the identity function: (∀ i ∈ S) 1S(i) = i. It should be pretty obvious that for any f : S → T, f º 1S = f; and 1T º f = f.

Partially Ordered Sets

Partially ordered sets (that is, sets that have a “<=" operator) can be described as a category, usually called PoSet. The objects are the partially ordered sets; the arrows are monotonic functions (a function f is monotonic if (∀ x,y &isin domain(x)) x <= y ⇒ f(x) <= f(y).). Like regular sets, º is function composition.

It’s pretty easy to show the associativity and identity properties; it’s basically the same as for sets, except that we need to show that º preserves the monotonicity property. And that’s not very hard:

  • Suppose we have arrows f : A → B, g : B → C. We know that f and g are monotonic
    functions.

  • Now, for any pair of x and y in the domain of f, we know that if x <= y, then f(x) <= f(y).
  • Likewise, for any pair s,t in the domain of g, we know that if s <= t, then g(s) <= g(t).
  • Put those together: if x <= y, then f(x) <= f(y). f(x) and f(y) are in the domain of g, so if (f(x) <= f(y)) then we know g(f(x)) <= g(f(y)).

Groups as a Category

There is a category Grp where the objects are groups; group homomorphisms are arrows. Homomorphisms are structure-preserving functions between sets; so function composition of those structure-preserving functions is the composition operator º. The proof that function composition preserves structure is pretty much the same as the proof we just ran through for partially ordered sets.

Once you have groups as a category, then you can do something very cool. If groups are a category, then functors over groups are symmetric transformations. Walk it through, and you’ll see that it fits. What took me a week of writing to be able to explain when I was talking about group theory can be stated in one sentence using the language of category theory. That’s a perfect example of why cat theory is useful: it lets you say some very important, very complicated things in very simple ways.

Miscellaneous Comments

There’ve been a couple of questions about from category theory skeptics in the comments. Please don’t think I’m ignoring you. This stuff is confusing enough for most people (me included) that I want to take it slowly, just a little bit at a time, to give readers an opportunity to digest each bit before going on to the next. I promise that I’ll answer your questions eventually!

Friday Random Ten

What kind of music does a math geek listen to?

  1. Capercaille: Who will raise their voice?. Traditional celtic folk music. Very beautiful song.
  2. Seamus Egan: Weep Not for the Memories. Mostly traditional Irish music, by a bizzarely talented multi-instrumentalist. Seamus Egan is one of the best Irish flutists in the world; but he also manages to play great tenor banjo, tenor guitar, six-string guitar, electric guitar, bohran, and keyboards.
  3. Gentle Giant: Experience. Gentle Giant is 70s progressive stuff, with heavy influence from early madrigal singing. Wierd, but incredibly cool.
  4. Tony Trischka Band: Steam/Foam of the Ancient Lake. Tony is my former banjo teacher. He’s also the guy who taught Bela Fleck to play Jazz. I have a very hard time deciding who I like better: Tony or Bela. They both do things with the banjo that knock my socks off. I think Bela gets a bit too much credit: not that he’s not spectacularly talented and creating; but he often gets credit for single-handedly redefining the banjo as an instrument, when Tony deserves a big share of the credit. This is a track off of the first album by Tony’s latest band. It’s great – I highly recommend the TTB to anyone.
  5. Trouth Fishing in America: Lullaby. TFiA is an incredibly great two-man folk band. They do both adult music, and music oriented towards children. Both are brilliant. Lullaby is, quite simple, one of the most beautifully perfect lullabies that I’ve ever heard. One of the two guys in TFiA, Ezra Idlet, is also somewhat famous for building a treehouse – not a kids treehouse, literally a treehouse: running water, electricity, central heating, etc. His house is a treehouse.
  6. Kind Crimson: B’Boom. A Bruford track off of one of Crimson’s recent albums. What more needs to be said?
  7. Dirty Three: Amy. The Dirty Three are something that they call a “post-rock ensemble”. All I can say is, it’s brilliant, amazing, fantastic music that I don’t know how to describe.
  8. Broadside Electric: Pastures of Plenty. Broadside is a Philadelpha based band that plays electrified folk. This is their take on an old folk track.
  9. Marillion: Ocean Cloud. Marillion is one of my favorite bands. They’re a neo-progressive group that started out as a Genesis cover band. Ocean Cloud is a long track off of their most recent album. It’s an amazing piece of work.
  10. Martin Hayes: Lucy Farr’s. Martin is a very traditional Irish fiddler. One of the really great things about him is that he’s really traditional. He doesn’t push the music to be ultrafast or showy; he takes it at speed that it was traditionally played, that you could dance to. It’s wonderful to hear the traditional tunes played right, without being over-adorned, over-accellerated, or otherwise mangled in the name of commericalism and ego.

Interesting mix today, all great stuff.

Diagrams in Category Theory

One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you’re talking about in a category, and then using category diagrams to express the idea that you want to get accross.

A category diagram is a directed graph, where the nodes are objects from a category, and the edges are morphisms. Category theorists say that a graph commutes if, for any two paths through arrows in the diagram from node A to node B, the composition of all edges from the first path is equal to the composition of all edges from the second path.

As usual, an example will make that clearer.
cat-assoc.jpg

This diagram is a way of expression the associativy property of morphisms: f º (g º h) = (f º g) º h. The way that the diagram illustrates this is: (g º h) is the morphism from A to C. When we compose that with f, we wind up at D. Alternatively, (f º g) is the arrow from B to D; if we compose that with H, we wind up at D. The two paths: f º (A → C), and (B → D) &ordm h are both paths from A to D, therefore if the diagram commutes, they must be equal.

Let’s look at one more diagram, which we’ll use to define an interesting concept, the principal morphism between two objects. The principle morphism is a single arrow from A to B, and any composition of morphisms that goes from A to B will end up being equivalent to it.

In diagram form, a morphism m is principle if (∀ x : A → A) (∀ y : A → B), the following diagram commutes.
cat-principal.jpg

In words, this says that f is a principal morphism if for every endomorphic arrow x, and for every arrow y from A to B, f is is the result of composing x and y. There’s also something interesting about this diagram that you should notice: A appears twice in the diagram! It’s the same object; we just draw it in two places to make the commutation pattern easier to see. A single object can appear in a diagram as many times as you want to to make the pattern of commutation easy to see. When you’re looking at a diagram, you need to be a bit careful to read the labels to make sure you know what it means. (This paragraph was corrected after a commenter pointed out a really silly error; I originally said “any identity arrow”, not “any endomorphic arrow”.)

One more definition by diagram: x and y are a retraction pair, and A is a retract of B (written A < B) if the following diagram commutes:
cat-retract.jpg

That is, x : A → B, and y : B → A are a retraction pair if y º x = 1A.

Why so many languages? Programming languages, Computation, and Math.

Back at my old digs last week, I put up a post about programming languages and types. It produced an interesting discussion, which ended up shifting topics a bit, and leading to a very interesting question from one of the posters, and since the answer actually really does involve math, I’m going to pop it up to the front page here.

In the discussion, I argued that programmers should know and use many different programming languages; and that that’s not just a statement about todays programming languages, but something that I think will always be true: that there will always be good reasons for having and using more than one programming language.

One of the posters was quite surprised by this, and wanted to know why I didn’t think that it was possible, at least in principle, to design a single ideal programming language that would be good for any program that I needed to write.

It’s a good question, which I think connects very naturally into the underlying mathematics of computation. As I discussed back on the old blog, one of the very fundamental facts concerning computation is the Church-Turing thesis: that computation has a fundamental limit, and that there are many different ways of performing mechanical computation, but ultimately, they all end up with the same fundamental capabilities. Any computing system that reaches that limit is called an effective computing system (ECS). Anything that one ECS can do, any other ECS can do too. That doesn’t mean that they’re all identical. A given computation can be really easy to understand when described in terms of one ECS, and horribly difficult in another. For example, sorting a list of numbers is pretty easy to understand if it’s written in lambda calculus; but it’s a nightmare written for a Minsky machine; and it’s somewhere darned close to impossible for a human to understand written out as a cell-grid for Conway’s life.

How does this connect back to programming languages? Well, what is a programming language really? From a theoretical point of view, it’s a language for specifying a computing machine. Each program is a description of a specific computing machine. The language is a particular way of expressing the computing machine that you want to build. Underlying each programming language is an effective computing system: a fundamental model of how to perform computations. That’s the real fundamental difference between programming languages: what fundamental way of representing computation underlies the language.

Assuming that you’re looking at a good programming language, it’s good for a particular task when that task is easy to describe in terms of the fundamental ECS underlying the language; it’s bad for a task when that task is not easy to describe in terms of the languages underlying ECS.

If a language has a single ECS underlying it, then there are tasks that it’s good for, and tasks that it’s not so good for. If you try to smoosh multiple ECSs together under the covers of one language, then you don’t really have one language: when you’re writing for a vonNeumann machine, you’re really using one language, and when you’re writing for Actors, you’re using another. You’ve just forced two languages together into the same framework – but they’re still two languages, and you’ve probably compromised the quality of expression of those two languages by forcing them together.

The beauty of math; the humor of stupidity.

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