Building Structure in Category Theory: Definitions to Build On

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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, I showed how you can use category theory to describe the basic idea of symmetry and group actions. Symmetry is, basically, an immunity to transformation – that is, a kind of structural property of an object or system where applying some kind of transformation to that object doesn’t change the object in any detectable way. The beauty of category theory is that it makes that definition much simpler.

Symmetry transformations are just the tip of the iceberg of the kinds of structural things we can talk about using categories. Category theory lets you build up pretty much any mathematical construct that you’d like to study, and describe transformations on it in terms of functors. In fact, you can even look at the underlying conceptual structure of category theory using category theory itself, by creating a category in which categories are objects, and functors are the arrows between categories.

So 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. The original inventor of category theory said that natural transformations were the real point of category theory – they’re what he wanted to study.

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.

Of course, this being a discussion of category theory, we can’t get any further without some definitions. To get to some of the interesting material that involves things like natural transformations, we need to know about a bunch of standard constructions: initial and final objects, products, exponentials… Then we’ll use those basic constructs to build some really fascinating constructs. That’s where things will get really fun.

So let’s start with initial and finial objects.

An initial object is a pretty simple idea: it’s an object with exactly one arrow to each of the other objects in the category. To be formal, given a category C, an object o \in Obj(C) is an initial object if and only if \forall b \in Obj(c): \exists_1 f: o \rightarrow b \in Mor(C). We generally write 0_c for the initial object in a category. Similarly, there’s a dual concept of a terminal object 1_c, which is object for which there’s exactly one arrow from every object in the category to 1_c.

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 \rightarrow b, where \exists g : b \rightarrow a) such that f \circ g = 1_b and g \circ f = 1_a. 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.

Now, let’s move on to categorical products. Categorical products define the product of two objects in a category. The basic concept is simple – it’s a generalization of 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 \in Obj(C), the categorical product a times b consists of:

  • An object p, often written a \times b;
  • two arrows p_a and p_b, where p \in Obj(C), p_a : p \rightarrow a, and p_b : p \rightarrow b.
  • a “pairing” operation, which for every object c \in C, maps the pair of arrows f : c \rightarrow a and
    g : c \rightarrow b to an arrow \langle f,g \rangle : c \rightarrow a\times b, where \langle f, g \rangle has the
    following properties:

    1. p_a \circ \langle f,g \rangle = f
    2. p_b \circ \langle f,g \rangle = g
    3. \forall h : c \rightarrow a \times b: \langle p_a \circ h, p_b \circ h \rangle = 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:

One important thing to understand is that categorical products do not have to exist. This definition doen not say that given any two objects a and b, that a times b is a member of the category. What it says is what the categorical product
looks like if it exists. If, for a given pair a and b of objects, there is an object that meets this definition, then the product of a and b exists in the category. If not, it doesn’t. For many categories, the products don’t exist for some or even all of the objects in the category. But as we’ll see later, the categories for which the products do exist have some really interesting properties.

Second Law Silliness from Sewell

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So, via Panda’s Thumb, I hear that Granville Sewell is up to his old hijinks. Sewell is a classic creationist crackpot, who’s known for two things.

First, he’s known for chronically recycling the old “second law of thermodynamics” garbage. And second, he’s known for building arguments based on “thought experiments” – where instead of doing experiments, he just makes up the experiments and the results.

The second-law crankery is really annoying. It’s one of the oldest creationist pseudo-scientific schticks around, and it’s such a terrible argument. It’s also a sort-of pet peeve of mine, because I hate the way that people generally respond to it. It’s not that the common response is wrong – but rather that the common responses focus on one error, while neglecting to point out that there are many deeper issues with it.

In case you’ve been hiding under a rock, the creationist argument is basically:

  1. The second law of thermodynamics says that disorder always increases.
  2. Evolution produces highly-ordered complexity via a natural process.
  3. Therefore, evolution must be impossible, because you can’t create order.

The first problem with this argument is very simple. The second law of thermodynamics does not say that disorder always increases. It’s a classic example of my old maxim: the worst math is no math. The second law of thermodynamics doesn’t say anything as fuzzy as “you can’t create order”. It’s a precise, mathematical statement. The second law of thermodynamics says that in a closed system:

Delta S geq int frac{delta Q}{T}

where:

  1. S is the entropy in a system,
  2. Q is the amount of heat transferred in an interaction, and
  3. T is the temperature of the system.

Translated into english, that basically says that in any interaction that involves the
transfer of heat, the entropy of the system cannot possible be reduced. Other ways of saying it include “There is no possible process whose sole result is the transfer of heat from a cooler body to a warmer one”; or “No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.”

Note well – there is no mention of “chaos” or “disorder” in these statements: The second law is a statement about the way that energy can be used. It basically says that when
you try to use energy, some of that energy is inevitably lost in the process of using it.

Talking about “chaos”, “order”, “disorder” – those are all metaphors. Entropy is a difficult concept. It doesn’t really have a particularly good intuitive meaning. It means something like “energy lost into forms that can’t be used to do work” – but that’s still a poor attempt to capture it in metaphor. The reason that people use order and disorder comes from a way of thinking about energy: if I can extract energy from burning gasoline to spin the wheels of my car, the process of spinning my wheels is very organized – it’s something that I can see as a structured application of energy – or, stretching the metaphor a bit, the energy that spins the wheels in structured. On the other hand, the “waste” from burning the gas – the heating of the engine parts, the energy caught in the warmth of the exhaust – that’s just random and useless. It’s “chaotic”.

So when a creationist says that the second law of thermodynamics says you can’t create order, they’re full of shit. The second law doesn’t say that – not in any shape or form. You don’t need to get into the whole “open system/closed system” stuff to dispute it; it simply doesn’t say what they claim it says.

But let’s not stop there. Even if you accept that the mathematical statement of the second law really did say that chaos always increases, that still has nothing to do with evolution. Look back at the equation. What it says is that in a closed system, in any interaction, the total entropy must increase. Even if you accept that entropy means chaos, all that it says is that in any interaction, the total entropy must increase.

It doesn’t say that you can’t create order. It says that the cumulative end result of any interaction must increase entropy. Want to build a house? Of course you can do it without violating the second law. But to build that house, you need to cut down trees, dig holes, lay foundations, cut wood, pour concrete, put things together. All of those things use a lot of energy. And in each minute interaction, you’re expending energy in ways that increase entropy. If the creationist interpretation of the second law were true, you couldn’t build a house, because building a house involves creating something structured – creating order.

Similarly, if you look at a living cell, it does a whole lot of highly ordered, highly structured things. In order to do those things, it uses energy. And in the process of using that energy, it creates entropy. In terms of order and chaos, the cell uses energy to create order, but in the process of doing so it creates wastes – waste heat, and waste chemicals. It converts high-energy structured molecules into lower-energy molecules, converting things with energetic structure to things without. Look at all of the waste that’s produced by a living cell, and you’ll find that it does produce a net increase in entropy. Once again, if the creationists were right, then you wouldn’t need to worry about whether evolution was possible under thermodynamics – because life wouldn’t be possible.

In fact, if the creationists were right, the existence of planets, stars, and galaxies wouldn’t be possible – because a galaxy full of stars with planets is far less chaotic than loose cloud of hydrogen.

Once again, we don’t even need to consider the whole closed system/open system distinction, because even if we treat earth as a closed system, their arguments are wrong. Life doesn’t really defy the laws of thermodynamics – it produces entropy exactly as it should.

But the creationist second-law argument is even worse than that.

The second-law argument is that the fact that DNA “encodes information”, and that the amount of information “encoded” in DNA increases as a result of the evolutionary process means that evolution violates the second law.

This absolutely doesn’t require bringing in any open/closed system discussions. Doing that is just a distraction which allows the creationist to sneak their real argument underneath.

The real point is: DNA is a highly structured molecule. No disagreement there. But so what? In the life of an organism, there are virtually un-countable numbers of energetic interactions, all of which result in a net increase in the amount of entropy. Why on earth would adding a bunch of links to a DNA chain completely outweigh those? In fact, changing the DNA of an organism is just another entropy increasing event. The chemical processes in the cell that create DNA strands consume energy, and use that energy to produce molecules like DNA, producing entropy along the way, just like pretty much every other chemical process in the universe.

The creationist argument relies on a bunch of sloppy handwaves: “entropy” is disorder; “you can’t create order”, “DNA is ordered”. In fact, evolution has no problem with respect to entropy: one way of viewing evolution is that it’s a process of creating ever more effective entropy-generators.

Now we can get to Sewell and his arguments, and you can see how perfectly they match what I’ve been talking about.

Imagine a high school science teacher renting a video showing a tornado sweeping through a town, turning houses and cars into rubble. When she attempts to show it to her students, she accidentally runs the video backward. As Ford predicts, the students laugh and say, the video is going backwards! The teacher doesn’t want to admit her mistake, so she says: “No, the video is not really going backward. It only looks like it is because it appears that the second law is being violated. And of course entropy is decreasing in this video, but tornados derive their power from the sun, and the increase in entropy on the sun is far greater than the decrease seen on this video, so there is no conflict with the second law.” “In fact,” the teacher continues, “meteorologists can explain everything that is happening in this video,” and she proceeds to give some long, detailed, hastily improvised scientific theories on how tornados, under the right conditions, really can construct houses and cars. At the end of the explanation, one student says, “I don’t want to argue with scientists, but wouldn’t it be a lot easier to explain if you ran the video the other way?”

Now imagine a professor describing the final project for students in his evolutionary biology class. “Here are two pictures,” he says.

“One is a drawing of what the Earth must have looked like soon after it formed. The other is a picture of New York City today, with tall buildings full of intelligent humans, computers, TV sets and telephones, with libraries full of science texts and novels, and jet airplanes flying overhead. Your assignment is to explain how we got from picture one to picture two, and why this did not violate the second law of thermodynamics. You should explain that 3 or 4 billion years ago a collection of atoms formed by pure chance that was able to duplicate itself, and these complex collections of atoms were able to pass their complex structures on to their descendants generation after generation, even correcting errors. Explain how, over a very long time, the accumulation of genetic accidents resulted in greater and greater information content in the DNA of these more and more complicated collections of atoms, and how eventually something called “intelligence” allowed some of these collections of atoms to design buildings and computers and TV sets, and write encyclopedias and science texts. But be sure to point out that while none of this would have been possible in an isolated system, the Earth is an open system, and entropy can decrease in an open system as long as the decreases are compensated by increases outside the system. Energy from the sun is what made all of this possible, and while the origin and evolution of life may have resulted in some small decrease in entropy here, the increase in entropy on the sun easily compensates this tiny decrease. The sun should play a central role in your essay.”

When one student turns in his essay some days later, he has written,

“A few years after picture one was taken, the sun exploded into a supernova, all humans and other animals died, their bodies decayed, and their cells decomposed into simple organic and inorganic compounds. Most of the buildings collapsed immediately into rubble, those that didn’t, crumbled eventually. Most of the computers and TV sets inside were smashed into scrap metal, even those that weren’t, gradually turned into piles of rust, most of the books in the libraries burned up, the rest rotted over time, and you can see see the result in picture two.”

The professor says, “You have switched the pictures!” “I know,” says the student. “But it was so much easier to explain that way.”

Evolution is a movie running backward, that is what makes it so different from other phenomena in our universe, and why it demands a very different sort of explanation.

This is a perfect example of both of Sewell’s usual techniques.

First, the essential argument here is rubbish. It’s the usual “second-law means that you can’t create order”, even though that’s not what it says, followed by a rather shallow and pointless response to the open/closed system stuff.

And the second part is what makes Sewell Sewell. He can’t actually make his own arguments. No, that’s much too hard. So he creates fake people, and plays out a story using his fake people and having them make fake arguments, and then uses the people in his story to illustrate his argument. It’s a technique that I haven’t seen used so consistency since I read Ayn Rand in high school.

The Annoying CTMU Thread

I had to close down the original comment thread discussing my rant about the crankery of Chris Langan’s CTMU.

The cost in server time to repeatedly retrieve that massive comment thread was getting excessive. All of Scientopia’s server costs are still coming out of my pocket, and I can’t afford a higher server bill. Since the server usage was approaching the point at which we’d need to up the resources (and thus, the bill), something had to be done.

So I shut down comments on that post. Interested parties are welcome to continue the discussion in the comment-thread under this post.

Free Energy Crankery and Negative Mass Nonsense

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I’ve got a couple of pet peeves.

As the author of this blog, the obvious one is bad math. And as I always say, the worst math is no math.

Another pet peeve of mine is free energy. Energy is, obviously, a hugely important thing to our society. And how we’re going to get the energy we need is a really serious problem – almost certainly the biggest problem that we face. Even if you’ve convinced yourself that global warming isn’t an issue, energy is a major problem. There’s only so much coal and oil that we can dig up – someday, we’re going to run out.

But there are tons of frauds out there who’ve created fake machines that they claim you can magically get energy from. And there are tons of cranks who are all-too-ready to believe them.

Take, for example, Bob Koontz.

Koontz is a guy with an actual physics background – he got his PhD at the University of Maryland. I’m very skeptical that he’s actually stupid enough to believe in what he’s selling – but nonetheless, he’s made a donation-based business out of selling his “free energy” theory.

So what’s his supposed theory?

It sounds impossible, but it isn’t. It is possible to obtain an unlimited amount of energy from devices which essentially only require that they be charged up with negative mass electrons and negative mass positrons. Any physicist should be able to convince himself of this in a matter of minutes. It really is simple: While ordinary positive mass electrons in a circuit consume power, negative mass electrons generate power. Why is that? For negative mass electrons and negative mass positrons, Newton’s second law, F = ma becomes F = -ma.

But acquiring negative mass electrons and negative mass electrons is not quite as simple as it sounds. They are exotic particles that many physicists may even doubt exist. But they do exist. I am convinced of this — for good reasons.

The Law of Energy Conservation

The law of energy conservation tells us that the total energy of a closed system is constant. Therefore, if such a system has an increase in positive energy, there must be an increase in negative energy. The total energy stays constant.

When you drop an object in the earth’s gravitational field, the object gains negative gravitational potential energy as it falls — with that increase in negative energy being balanced by an increase of positive energy of motion. But the object does not lose or gain total energy as it falls. It gains kinetic energy while it gains an equal amount of negative gravitational energy.

How could we have free energy? If we gain positive energy, we must also generate negative energy in exactly the same amount. That will “conserve energy,” as physicists say. In application, in the field of “free energy,” that means generating negative energy photons and other negative energy particles while we get the positive energy we are seeking. What is the problem, then? The problem involves generating the negative energy particles.

a

So… there are, supposedly, “negative energy” particles that correspond to electrons and positrons. These particles have never been observed, and under normal circumstances, they have no effect on any observable phenomenon.

But, we’re supposed to believe, it really exists. And it means that we can get free energy without violating the conservation of energy – because the creation of an equal amount of invisible, undetectable, effectless negative energy balances out whatever positive energy we create.

So what is negative energy?

That’s where the bad math comes in. Here’s his explanation:

When Paul Dirac, the Nobel prize-winning physicist was developing the first form of relativistic quantum mechanics he found it necessary to introduce the concept of negative mass electrons. This subsequently led Dirac to develop the idea that a hole in a sea of negative mass electrons corresponded to a positron, otherwise known as an antielectron. Some years later the positron was observed and Dirac won the Nobel prize.

Subsequent to the above, there appears to have been no experimental search for these negative mass particles. Whether or not negative mass electrons and negative mass positrons exist is thus a question to which we do not yet have an answer. However, if these particles do exist, their unusual properties could be exploited to produce unlimited amounts of energy — as negative mass electrons and negative mass positrons, when employed in a circuit, produce energy rather than consume it. Newton’s 2nd law F = ma becomes F = – ma and that explains why negative mass electrons and negative mass positrons produce energy rather than consume it. I believe that any good physicist should be able to see this quite quickly.

The following paragraph is actually wrong. There is such a thing as relativistic quantum mechanics. QM and special relativity are compatible, and the relativistic QM fits is at that intersection. General relativity and QM remains an unsolved problem, as discussed below. I’m leaving the original paragraph, because it seems dishonest to just delete it, like I was pretending that I never screwed up.

There is no such thing as relativistic quantum mechanics. One of the great research areas of modern physics is the attempt to figure out how to unify quantum mechanics and relativity. Many people have tried to find a unifying formulation, but no one has yet succeeded. There is no theory of relativistic QM.

It’s actually a fascinating subject. General relativity seems to be true: every test that we can dream up confirms GR. And quantum mechanics also appears to be true: every test that we can dream up confirms the theory of quantum mechanics. And yet, the two are not compatible.

No one has been able to solve this problem – not Dirac, not anyone.

Even within the Dirac bit… there is a clever bit of slight-of-hand. He starts by saying that Dirac proposed that there were “negative mass” electrons. Dirac did propose something like that – but the proposal was within the frame of mathematics. Without knowing about the existence of the positron, he worked through the implications of relativity, and would up with a model which could be interpreted as a sea of “negative mass” electrons with holes in it. The holes are positrons.

To get a sense of what this means, it’s useful to pull out a metaphor. In semiconductor physics, when you’re trying to describe the behavior of semiconductors, it’s often useful to talk about things backwards. Instead of talking about how the electrons move through a semiconductor, you can talk about how electron holes move. An electron hole is a “gap” where an electron could move. Instead of an electron moving from A to B, you can talk about an electron hole moving from B to A.

The Dirac derivation is a similar thing. The real particle is the positron. But for some purposes, it’s easier to discuss it backwards: assume that all of space is packed, saturated, with “negative mass” electrons. But there are holes moving through that space. A hole in a “negative mass”, negatively charged field is equivalent to a particle with positive mass and positive charge in an empty, uncharged space – a positron.

The catch is that you need to pick your abstraction. If you want to use the space-saturated-with-negative-mass model, then the positron doesn’t exist. You’re looking at a model in which there is no positron – there is just a gap in the space of negative-mass particles. If you want to use the model with a particle called a positron, then the negative mass particles don’t exist.

So why haven’t we been searching for negative-mass particles? Because they don’t exist. That is, we’ve chosen the model of reality which says that the positron is a real particle. Or to be slightly more precise: we have a very good mathematical model of many aspects of reality. In that model, we can choose to interpret it as either a model in which the positive-mass particles really exist and the negative-mass particles exist only as an absence of particles; or we can interpret it as saying that the negative-mass particles exist, and the positive mass ones exist only as an absence of negative-mass particles. In either case, that model provides an extremely good description of what we observe about reality. But that model does not predict that both the positive and negative mass particles both really exist in any meaningful sense. By observing and calculating the properties of the positive mass particles, we adopt the interpretation that positive mass particles really exist. Every observation that we make of the properties of positive mass particles is implicitly an observation of the properties of negative-mass particles. The two interpretations are mathematical duals.

Looking at his background and at at other things on his site, I think that Koontz is, probably, a fraud. He’s not dumb enough to believe this. But he’s smart enough to realize that there are lots of other people who are dumb enough to believe it. Koontz has no problem with pandering to them in the name of his own profit. What finally convinced me of that was his UFO-sighting claim here. Absolutely pathetic.

Facts vs Beliefs

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One of the things about current politics that continually astonishes me is the profound lack of respect for reality demonstrated by so many of the people who want to be in charge of our governments.

Personally, I’m very much a liberal. I lean way towards the left-end of the political spectrum. But for the purposes of this discussion, that’s irrelevant. I’m not talking about whether people are proposing the right policy, or the right politics. What I’m concerned with is the way that the don’t seem to accept the fact that there are facts. Not everything is a matter of opinion. Some things are just undeniable facts, and you need to deal with them as they are. The fact that you don’t like them is just irrelevant. As the old saying goes, you’re entitled to your own opinion, but you’re not entitled to your own facts.

I saw a particularly vivid example of this last week, but didn’t have a chance to write it up until today. Rick Perry was presenting his proposal for how to address the problems of the American economy, particularly the dreadfully high unemployment rate. He claims that his policy will, if implemented, create 2.5 million jobs over the next four years.

The problem with that, as a proposal, is that in America, due to population growth, just to break even in employment, we need to add 200,000 jobs per month – that’s how fast the pool of employable people is growing. So we need to add over two million jobs per year just to keep unemployment from rising. In other words, Perry is proposing a policy that will, according to his (probably optimistic, if he’s a typical politician) estimate, result in increasing unemployment.

This is, obviously, bad.

But here’s where he goes completely off the rails.

Chris Wallace: “But how do you answer this question? Two and a half million jobs doesn’t even keep pace with population growth. Our unemployment rate would increase under this goal.

Rick Perry: “I don’t believe that for a minute. It’s just absolutely false on its face. Americans will get back to work.”

That’s just blatant, stupid idiocy.

The employable population is growing. This is not something debatable. This is not something that you get to choose to believe or not to believe. This is just reality.

If you add 2.5 million jobs, and the population of employable workers seeking jobs grows by 4 million people, then the unemployment rate will get worse. That’s simple arithmetic. It’s not politics, it’s not debatable, and it has nothing to do with what Rick Perry, or anyone else, believes. It’s a simple fact.

The fact that a candidate for president can just wave his hands and deny reality – and that that isn’t treated as a disqualifying error – is simply shocking.

Yet Another Cantor Crank

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I get a fair bit of mail from crackpots. The category that I find most annoying is the Cantor cranks. Over and over and over again, these losers send me their “proofs”.

What bugs me so much about this is how shallowly wrong they are.

What Cantor did was remarkably elegant. He showed that given anything that is claimed to be a one-to-one mapping between the set of integers and the set of real numbers (also sometimes described as an enumeration of the real numbers – the two terms are functionally equivalent), then here’s a simple procedure which will produce a real number that isn’t in included in that mapping – which shows that the mapping isn’t one-to-one.

The problem with the run-of-the-mill Cantor crank is that they never even try to actually address Cantor’s proof. They just say “look, here’s a mapping that works!”

So the entire disproof of their “refutation” of Cantor’s proof is… Cantor’s proof. They completely ignore the thing that they’re claiming to disprove.

I got another one of these this morning. It’s particularly annoying because he makes the same mistake as just about every other Cantor crank – but he also specifically points to one of my old posts where I rant about people who make exactly the same mistake as him.

To add insult to injury, the twit insisted on sending me PDF – and not just a PDF, but a bitmapped PDF – meaning that I can’t even copy text out of it. So I can’t give you a link; I’m not going to waste Scientopia’s bandwidth by putting it here for download; and I’m not going to re-type his complete text. But I’ll explain, in my own compact form, what he did.

It’s an old trick; for example, it’s ultimately not that different from what John Gabriel did. The only real novelty is that he does it in binary – which isn’t much of a novelty. This author calls it the “mirror method”. The idea is, in one column, write a list of the integers greater than 0. In the opposite column, write the mirror of that number, with the decimal (or, technically, binary) point in front of it:

Integer Real
0 0.0
1 0.1
10 0.01
11 0.11
100 0.001
101 0.101
110 0.011
111 0.111
1000 0.0001
… …

Extend that out to infinity, and, according to the author, the second column it’s a sequence of every possible real number, and the table is a complete mapping.

The problem is, it doesn’t work, for a remarkably simple reason.

There is no such thing as an integer whose representation requires an infinite number of digits. For every possible integer, its representation in binary has a fixed number of bits: for any integer N, it’s representation is no longer that lceil log_2(n) rceil. That’s always a finite integer.

But… we know that the set of real numbers includes numbers whose representation is infinitely long. so this enumeration won’t include them. Where does the square root of two fall in this list? It doesn’t: it can’t be written as a finite string in binary. Where is π? It’s nowhere; there’s no finite representation of π in binary.

The author claims that the novel property of his method is:

Cantor proved the impossibility of both our enumerations as follows: for any given enumeration like ours Cantor proposed his famous diagonal method to build the contra-sample, i.e., an element which is quasi omitted in this enumeration. Before now, everyone agreed that this element was really omitted as he couldn’t tell the ordinal number of this element in the give enumeration: now he can. So Cantor’s contra-sample doesn’t work.

This is, to put it mildly, bullshit.

First of all – he pretends that he’s actually addressing Cantor’s proof – only he really isn’t. Remember – what Cantor’s proof did was show you that, given any purported enumeration of the real numbers, that you could construct a real number that isn’t in that enumeration. So what our intrepid author did was say “Yeah, so, if you do Cantor’s procedure, and produce a number which isn’t in my enumeration, then I’ll tell you where that number actually occurred in our mapping. So Cantor is wrong.”

But that doesn’t actually address Cantor. Cantor’s construction specifically shows that the number it constructs can’t be in the enumeration – because the procedure specifically guarantees that it differs from every number in the enumeration in at least one digit. So it can’t be in the enumeration. If you can’t show a logical problem with Cantor’s construction, then any argument like the authors is, simply, a priori rubbish. It’s just handwaving.

But as I mentioned earlier, there’s an even deeper problem. Cantor’s method produces a number which has an infinitely long representation. So the earlier problem – that all integers have a finite representation – means that you don’t even need to resort to anything as complicated as Cantor to defeat this. If your enumeration doesn’t include any infinitely long fractional values, then it’s absolutely trivial to produce values that aren’t included: 1/3, 1/7, 1/9.

In short: stupid, dull, pointless; absolutely typical Cantor crankery.

Fun with Functors

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So far, we’ve looked at the minimal basics of categories: what they are, and how to categorize the kinds of arrows that exist in categories in terms of how they compose with other arrows. Just that much is already enlightening about the nature of category theory: the focus is always on composition.

But to get to really interesting stuff, we need to build up a bit more, so that we can look at more interesting constructs. So now, we’re going to look at functors. Functors are one of the most fundamental constructions in category theory: they give us the ability to create multi-level constructions.

What’s a functor? Well, it’s basically a structure-preserving mapping between categories. So what does that actually mean? Let’s be a bit formal:

A functor F from a category C to a category D is a mapping from C to D that:

  • Maps each member m in Obj(C) to an object F(m) in Obj(D).
  • Maps each arrow a : x rightarrow y in Mor(C) to an arrow F(a) : F(x) rightarrow F(y), where:
    • forall o in Obj(C): F(1_o) = 1_{F(o)}. (Identity is preserved by the functor mapping of morphisms.)
    • forall m,n in Mor(C): F(n circ o) = F(n) circ F(o). (Commutativity is preserved by the Functor mapping of morphisms.)

Note: The original version of this post contained a major typo. In the second condition on functors, the “n” and the “o” were reversed. With them in this direction, the definition is actually the definition of something called a covariant functor. Alas, I can’t even pretend that I mixed up covariant and contravariant functors; the error wasn’t nearly so intelligent. I just accidentally reversed the symbols, and the result happened to make sense in the wrong way.

That’s the standard textbook gunk for defining a functor. But if you look back at the original definition of a category, you should notice that this looks familiar. In fact, it’s almost identical to the definition of the necessary properties of arrows!

We can make functors much easier to understand by talking about them in the language of categories themselves. Functors are really nothing but morphisms – they’re morphisms in a category of categories.

There’s a kind of category, called a small category. (I happen to dislike the term “small” category, but I don’t get a say!) A small category is a category whose collections of objects and arrows are sets, not proper classes.

(As a quick reminder: in set theory, a class is a collection of sets that can be defined by a non-paradoxical property that all of its members share. Some classes are sets of sets; some classes are not sets; they lack some of the required properties of sets – but still, the class is a collection with a well-defined, non-paradoxical, unambiguous property. If a class isn’t a set of sets, but just a collection that isn’t a set, then it’s called a proper class.)

Any category whose collections of objects and arrows are sets, not proper classes, are called small categories. Small categories are, basically, categories that are well-behaved – meaning that their collections of objects and arrows don’t have any of the obnoxious properties that would prevent them from being sets.

The small categories are, quite beautifully, the objects of a category called Cat. (For some reason, category theorists like three-letter labels.) The arrows of Cat are all functors – functors really just morphisms between categories. Once you wrap you head around that, then the meaning of a functor, and the meaning of a structure-preserving transformation become extremely easy to understand.

Functors come up over and over again, all over mathematics. They’re an amazingly useful notion. I was looking for a list of examples of things that you can describe using functors, and found a really wonderful list on wikipedia.. I highly recommend following that link and taking a look at the list. I’ll just mention one particularly interesting example: groups and group actions.

If you’ve been reading GM/BM for a very long time, you’ll remember my posts on group theory. In a very important sense, the entire point of group theory is to study symmetry. But working from a set theoretic base, it takes a lot of work to get to the point where you can actually define symmetry. It took many posts to build up the structure – not to present set theory, but just to present the set theoretic constructs that you need to define what symmetry means, and how a symmetric transformation was nothing but a group action. Category theory makes that so much easier that it’s downright dazzling. Ready?

Every group can be represented as a category with a single object. A functor from the category of a group to the category of Sets is a group action on the set that is the target of the functor. Poof! Symmetry.

Since symmetry means structure-preserving transformation; and a functor is a structure preserving transformation – well, they’re almost the same thing. The functor is an even more general abstraction of that concept: group symmetry is just one particular case of a functor transformation. Once you get functors, understanding symmetry is easy. And so are lots of other things.

And of course, you can always carry these things further. There is a category of functors themselves; and notions which can be most easily understood in terms of functors operating on the category of functors!

This last bit should make it clear why category theory is affectionately known as abstract nonsense. Category theory operates at a level of abstraction where almost anything can be wrapped up in it; and once you’ve wrapped something up in a category, almost anything you can do with it can itself be wrapped up as a category – levels upon levels, categories of categories, categories of functors on categories of functors on categories, ad infinitum. And yet, it makes sense. It captures a useful, comprehensible notion. All that abstraction, to the point where it seems like nothing could possibly come out of it. And then out pops a piece of beautiful crystal. It’s really remarkable.

Category Diagrams

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One of the most unusual things about category theory that I really love is category diagrams. In category theory, many things that would be expressed as equations or complex formulae in most mathematical formalisms can be presented as diagrams in category theory. If you are, like me, a very visual thinker, than category diagrams can present information in a remarkably clear form – and the categorical form of many statements of proofs can be much clearer than the alternatives because it can be presented in diagram form.

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. (But it’s important to note that you do need to be careful here: merely because you can draw a diagram doesn’t mean that it necessarily commutes, just like being able to write an equation doesn’t mean that the equation is true! You do need to show that your diagram is correct and commutes.)

As usual, an example will make that clearer.

cat-assoc.jpg

This diagram is a way of expressing the associativy property of morphisms: f circ (g circ h) = (f circ g) circ h. The way that the diagram illustrates this is: g circ h is the morphism from A to C. When we compose that with f, we wind up at D. Alternatively, f circ g is the arrow from B to D; if we compose that with h, we wind up at D. The two paths: f circ (A rightarrow C) and (B rightarrow D) circ H are both paths from A to D, therefore if the diagram commutes, they must be equal. And the arrows on the diagram are all valid arrows, arranged in connections that do fit the rules of composition correctly, so the diagram does commute.

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 such that 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 forall x : A rightarrow A, forall y: A rightarrow 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.

One more definition by diagram: (x, y) is a 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 rightarrow B, y: B rightarrow A are a retraction pair if y circ x = 1_A.

I get mail: Brown's Gas and Perpetual Motion

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In the past, I’ve written about free-energy cranks like Tom Bearden, and I’ve made several allusions to the Brown’s gas” crankpots. But I’ve never actually written in any detail about the latter.

Brown’s gas is a term used primarily by cranks for oxyhydrogen gas. Oxyhydrogen is a mixture of hydrogen and oxygen in a two-to-one molar ratio; in other words, it’s exactly the product of electrolysis to break water molecules into hydrogen and oxygen. It’s used as the fuel for several kinds of torches and welders. It’s become a lot less common, because for most applications, it’s just not as practical as things like acetylene torches, TIG welders, etc.

But for free-energy cranks, it’s a panacea.

You see, the beautiful thing about Brown’s gas is that it burns very nicely, it can be compressed well enough to produce a very respectable energy density, and when you use it, its only exhaust gas is water. If you look at it naively, that makes it absolutely wonderful as a fuel.

The problem, of course, is that it costs energy to produce it. You need to pump energy into water to divide it into hydrogen and oxygen; and then you need to use more energy to compress it in order to make it useful. Still, there are serious people who are working hard on things like hydrogen fuel cell power sources for cars – because it is an attractive fuel. It’s just not a panacea.

But the cranks… Ah, the cranks. The cranks believe that if you just find the right way to burn it, then you can create a perfect source of free energy. You see, if you can just burn it so that it produces a teeny, tiny bit more energy being burned that it cost to produce, then you’ve got free energy. You just run an engine – it keeps dividing the water into hydrogen and oxygen, and then you burn it, producing more energy than you spent to divide it; and the only by-product is water vapor!

Of course, this doesn’t work. Thermodynamics fights back: you can’t get more energy out of recombining atoms of hydrogen and oxygen than you spent splitting molecules of water to get that hydrogen and oxygen. It’s very simple: there’s a certain amount of latent energy in that chemical bond. You need to pump in a certain amount of energy to break it – if I remember correctly, it’s around 142 Joules per gram of water. When you burn hydrogen and oxygen to produce water, you get exactly that amount of energy back. It’s a state transition – it’s the same distance up as it is back down. It’s like lifting a weight up a step on a staircase: it takes a certain amount of energy to move the weight up one step. When you drop it back down, it won’t produce more energy falling that you put in to lift it.

But the Brown’s gas people won’t let that stop them!

Here’s an email I recieved yesterday from a Brown’s gas fan, who noticed one of my old criticisms of it:

Hi Mark,

My name is Stefan, and I recently came across your analysis regarding split water technology to power vehicle. You are trying to proof that it makes no sense because it is against the physic low of energy conservation?

There is something I would like to ask you, if you could explain to me. What do you think about the sail boat zigzagging against the wind? Is it the classical example of perpetual motion?

If so, I believe that the energy conversion law is not always applicable, and even maybe wrong? Using for example resonance you can destroy each constructions with little force, the same I believe is with membrane HHO technology at molecular level?

Is it possible that we invented the law of impossibility known as the Energy Conservation Law and this way created such limitation? If you have some time please answer me what do you think about it? This World as you know is mostly unexplainable, and maybe we should learn more to better understand how exactly the Universe work?

The ignorance in this is absolutely astonishing. And it’s pretty typical of my experience with the Brown’s gas fans. They’re so woefully ignorant of simple math and physics.

Let’s start with his first question, about sailboat tacking. That’s got some interesting connections to my biggest botch on this blog, my fouled up debunking of the downwind-faster-than-the-wind vehicle.

The tacking sailboat is a really interesting problem. When you think about it naively, it seems like it shouldn’t be possible. If you let a leaf blow in the wind, it can’t possibly move faster than the wind. So how can a sailboat do it?

The anwser to that is that the sailboat isn’t a free body floating in the wind. It’s got a body and keel in the water, and a sail in the air. What it’s doing is exploiting that difference in motion between the water and the air, and extracting energy. Mathematically, the water behaves as a source of tension, resisting the pressure of the wind against the sail, and converting it into motion in a different direction. Lift the body of the sailboat out of the water, and it can’t do that anymore. Similarly, a boat can’t accelerate by “tacking” against the water current unless it has a sail. It needs the two parts in different domains; then it can, effectively, extract energy from the difference between the two. But the most important point about a tacking sailboat – more important than the details of the mechanism that it uses – is that there’s no energy being created. The sailboat is extracting kinetic energy from the wind, and converting it into kinetic energy in the boat. There’s no energy being created or destroyed – just moved around. Every bit of energy that the boat acquires (plus some extra) was removed from the wind.

So no, a sailboat isn’t an example of perpetual motion. It’s just a very typical example of moving energy around from one place to another. The sun heats the air/water/land; that creates wind; wind pushes the boat.

Similarly, he botches the resonance example.

Resonance is, similarly, a fascinating phenomenon, but it’s one that my correspondant totally fails to comprehend.

Resonance isn’t about a small amount of energy producing a large effect. It’s about how a small amount of energy applied over time can add up to a large amount of energy.

There is, again, no energy being created. The resonant system is not producing energy. A small amount of energy is not doing anything more than a small amount of energy can always do.

The difference is that in the right conditions, energy can add in interesting ways. Think of a spring with a weight hanging on the end. If you apply a small steady upward force on the weight, the spring will move upward a small distance. When you release the force, the weight will fall to slightly below its apparent start point, and then start to come back up. It will bounce up and down until friction stops it.

But now… at the moment when it hits its highest position, you give it another tiny push, then it will move a bit higher. Now it’s bounce distance will be longer. If every time, exactly as it hits its highest point, you give it another tiny push, then each cycle, it will move a little bit higher. And by repeatedly appyling tiny forces at the right time, the forces add up, and you get a lot of motion in the spring.

The key is, how much? And the answer is: take all of the pushes that you gave it, and add them up. The motion that you got from the resonant pattern is exactly the same as the motion you’d get if you applied the summed force all at once. (Or, actually, you’d get slightly more from the summed force; you lost some to friction in the resonant scenario.

Resonance can create absolutely amazing phenomena, where you can get results that are absolutely astonishing; where forces that really seem like they’re far to small to produce any result do something amazing. The famous example of this is the Tacoma Narrows bridge collapse, where the wind happened to blow just right to created a resonant vibration which tore the bridge apart:

But there’s no free energy there; no energy being created or destroyed.

So, Stefan… It’s always possible that we’re wrong about how physics work. It’s possible that conservation of energy isn’t a real law. It’s possible that the world might work in a way where conservation of energy just appears to be a law, and in fact, there are ways around it, and that we can use those ways to produce free energy. But people have been trying to do that for a very, very long time. We’ve been able to use our understanding of physics to do amazing things. We can accelerate particles up to nearly the speed of light and slam them together. We can shoot rockets into space. We can put machines and even people on other planets. We can produce energy by breaking atoms into pieces. We can build devices that flip switches billions of times per second, and use them to talk to each other! And we can predict, to within a tiny fraction of a fraction of the breadth of a hair how much energy it will take to do these things, and how much heat will be produced by doing them.

All of these things rely on a very precise description of how things work. If our understanding were off by the tiniest bit, none of these things could possibly work. So we have really good reasons to believe that our theories are, to a pretty great degree of certainty, accurate descriptions of how reality works. That doesn’t mean that we’re right – but it does mean that we’ve got a whole lot of evidence to support the idea that energy is always conserved.

On the side of the free energy folks: not one person has ever been able to demonstrate a mechanism that produces more energy than was put in to it. No one has ever been able to demonstrate any kind of free energy under controlled experimental conditions. No one has been able to produce a theory that describes how such a system could work that is consistent with observations of the real world.

People have been pushing Brown’s gas for decades. But they’ve never, every, not one single time, been able to actually demonstrate a working generator. No one has ever done it. No one has been able to build a car that actually works using Brown’s gas without an separate power source. No one has build a self-sustaining generator. No one has been able to produce any mathematical description of how Brown’s gas produces energy that is consistent with real-world observations.

So you’ve got two sides to the argument about Brown’s gas. On one side, you’ve got modern physics, which has reams and reams of evidence, precise theories that are confirmed by observation, and unbelievable numbers of inventions that rely on the precision of those theories. On the other side, you’ve got people who’ve never been able to to do a demonstration, who can’t describe how things work, who can’t explain why things appear to work the way that they appear, who have never been able to produce a single working invention…

Which side should we believe? Given the current evidence, the answer is obvious.

Category Intuition by Example

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The good thing about category theory is that it abstracts everything down to a couple of very simple, bare-bones concepts, and lets us play with those concepts in really fascinating ways. But the bad thing about category theory is that it abstracts everything down so much that it can be difficult to have any intuitive sense about what things actually mean.

We said that a category is, basically, a collection of objects connected by arrows, and a composition operation over the arrows. But what the heck is an object, and what is an arrow?

In some sense, the point of category theory is that it doesn’t matter. There’s a set of fundamental concepts that underly all composable mapping-like things, and category theory focuses in on those fundamental concepts, and what they mean.

But still, to understand category theory, you need to be able to get some kind of intuitive handle on what’s going on. And so, we’ll take a look at a few examples.

The Category Set

One of the easiest examples is the category Set. The objects of the category are, obviously, sets; the morphisms are functions between sets; and composition is (obviously) function composition.

That much is easy. Now, what about the special arrows? What do all of the epi, iso, endo, and auto arrows in the category of sets?

An monomorphism over the category of sets is an injective function – that is, a function which maps each value in the domain to a distinct value in the range. So you can always reverse the function: given a value in the range of the function, you can always say, specifically, exactly what value in the domain maps to it. In a monomorphism in the category set, there may be values in the target of the morphism that aren’t mapped to by any element of the range; but if a value of the range is mapped to, you know which value in the domain mapped to it.

The key property of the monomorphism is that it’s left-cancellative – that is, if f is a monomorphism, then we know that f circ g == f circ h is only true if g = h. We can cancel the left side of the composion. Why? Because we know that f maps each value in its domain to a unique value in its range. So f circ g and f circ h can only be the same if they map all of the same values in their domain to the same values in their range – that is, if they’re the same function.

An epimorphism is an onto function – that is, a mapping f from a set X to a set Y in which for every value y in Y, there’s some value in x in X such that f(x)=y. It’s a dual of the notion of a monomorphism; for each value in the range, there is at least one value in the domain that maps to it. But it’s not necessary that the domain values be unique – there can be multiple values in the domain that map onto a particular value in the range, but for every element of the domain, there’s always at least one value that maps to it. It’s right-cancellative in the same way that the monomorphism is left-cancellative.

What about iso-morphism? If you combine the ideas of mono-morphism and epi-morphism, what you end up with is a function where every member of the domain is mapped onto a unique value in the range, and every value in the range is mapped onto by at least one value: it’s a one-to-one function.

The Category Poset

Poset is the category of partially ordered sets. A partially ordered set is a set with a binary relation, le, which satisfies the following properties:

  1. Reflexivity: forall a: a le a
  2. Antisymmetry: if a le b land b le a then a = b.
  3. Transitivity: if a le b land b le c then a le c

In the category of partially ordered sets, the arrows are monotonic functions – that is, functions that preserve the partial ordering. So if x le y, then if f is monotonic, f(x) le f(y).

The arrow composition operation, circ, is still just function composition. In terms of meaning, it’s pretty much the same as it was for simple sets: a monomorphism is still a surjective function; etc. But in Poset it needs to be a surjective function that preserves the ordering relation.

We know that monotonic functions fit the associative and identity properties of category arrows – they’re still just functions, and monotonicity has no effect on those. But for posets, the composition is a bit tricky – we need to ensure that composition maintains the partial order. Fortunately, that’s easy.

  • Suppose we have arrows f : A rightarrow B, g : B rightarrow C. We know that f and g are monotonic functions.
  • So for any pair of x and y in the domain of f, we know that if x le y, then f(x) le f(y).
  • Likewise, for any pair s,t in the domain of g, we know that if s le t, then g(s) le g(t).
  • So we just put those together: if x le y, then f(x) le f(y). f(x) and f(y) are in the domain of g, so if f(x) le f(y) then we know g(f(x)) le g(f(y))

The category Grp

Our last example is the category of groups, which is called Grp. In this category, the objects are (obviously) groups. What are arrows? Group homomorphisms – that is, essentially, functions between sets that preserve the group structure.

What’s a homomorphism in this category? That’s a bit confusing, because we get stuck with some overloaded terminology. But it’s basically just a symmetry-preserving surjection – that is, it’s a reversable mapping from a group into a second group, where the group symmetry of the first group is mapped onto the group symmetry of the second by the arrow. You should be able to follow the meanings of the other kinds of morphisms by using similar reasoning.

These are all, of course, very simple examples. They’re all concrete categories, where the objects are (speaking very loosely) sets. There are categories – many very interesting ones – where the objects aren’t sets or even particularly set-like – but we’ll get to those later. For now, the ideas of objects as sets gives you an intuitive handle to grab onto.