Free Energy by Switching Cameras (Classic Repost)

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This is an edited repost of a classic. Back when I first put up the post about the genius theories of Engineer Borg, a commenter pointed me towards the website of a Dr. Tom Bearden. Dr. Bearden is a veritable renaissance man of crackpottery: he hasinvented a perfect free energy system which has been quashed by a conspiracy of governments and corporations; invented a cure for all major diseases (again hidden by the strenuous efforts of corporations and governments); demonstrated the flaw in relativity… You name it, Tom has done it!

Before I get to the details of that, let me give you a sense of the flavor of his site. Dr. Tom clearly believes that he is a genius of epic proportions, and that the entire world actually knows it. For example, he repeatedly talks about how his work was favorably reviewed by the National Science Foundation! Which means it’s brilliant! Only it was quashed by the Evil Government Conspiracy before he could demonstrate it! So I went looking for the supposed favorable review of his free-energy work. And I found it in his list of references, listed as “National Science Foundation letter favorably reviewing Bearden Paper”.

This looks interesting, right? A review from the NSF? So, click the link, and… The contents of that link consist of a scanned letter from the NSF replying to an email sent by Dr. Bearden, which consists of a basic standardized form letter inviting him to submit an actual proposal, and warning that he’d better include some proof that his perpetual motion machine really works, and an explanation of how.

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My Newest Flute, made of… Plastic?!

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This is rather off topic for GM/BM, but there’s a teeny bit of physics mixed in.

One of the things that I do for fun, other than writing this blog, is playing the flute. I don’t play the modern flute: I play traditional Irish music on the wooden flute. For traditional Irish music, you’re mostly playing tunes that were written for pipes, which aren’t chromatic – and as a result, for Irish music, you don’t actually need any keys. Just the main six finger holes are enough. I bought a really magnificent wooden flute, custom made by an amazing craftsman named Patrick Olwell.

But sometimes, I want to be able to play other stuff. So for a very long time, I’ve wanted a wooden flute with keys, a flute that could play chromatically so that I could play any kind of music I wanted. The problem is, a decent keyed wooden flute costs a fortune. They generally cost at least $4,000, and most of the good makers have a waiting list. For Pat Olwell, that waiting list is between three and seven years.

So for a very long time, I’ve been looking for a way of getting a keyed, chromatic wooden flute. I’ve bought four different antiques from Ebay, all of which needed lots of work to be playable, and none of which were really salvagable for chromatic playing – their keywork is just too messed up for me to fix.

I’d heard about M&E, a plastic flute made by a guy named Michael Cronnoly. His flutes are much less expensive, and they’ve got a very good reputation.

But… Plastic?

I’ve seen several acoustic studies that claim that the material the instrument is made of isn’t that important. In a wooden flute, the physics show that the head joint is the only part of the flute that really has a significant influence on its sound. But the head joint of a wooden flute is actually lined with metal. So the wood isn’t really having too much influence on the sound.

But…

The first flute I bought was a Dixon polymer. The thing is, frankly, a piece of junk. It’s incredibly heavy; the tone is mediocre at best; the embouchure hole is awful… It’s really not a great instrument. That’s my only prior experience with pseudo-wooden flutes, and it really wasn’t a good one.

Plus, I grew up playing the clarinet. There’s a similar argument about acoustic materials for clarinets. In a clarinet, the tone is formed in the mouthpiece and barrel: they determine how it will sound. Most people (including me) play on mouthpieces made of hard rubber or plastic – so the primary sound-producing piece of the instrument is plastic. The barrel of a wooden clarinet is (obviously) wood, so according to the physics/acoustics, that’s the only piece of wood that actually has any measurable acoustic effect. And the physics of this isn’t sloppy stuff put together by an instrument company trying to sell their plastic clarinets: to the limits of my ability to understand it, it’s good, solid stuff.

And yet, I’ve played a whole lot of clarinets, and by god, there’s nothing like a grenadilla wood clarinet. Even the best clarinet makers, even when I put my wooden barrel on a polymer body, it doesn’t sound the same. Of course, that’s subjective, and we humans are notorious for hearing what we want to hear in a subjective situation. And, by god, I’m a math geek. I’ve seen the math, and it’s correct.

But still, I really do believe that my wooden clarinet sounds better than any plastic I’ve ever played. So why? If the math says it shouldn’t, why does it? I’ve never been sure, but my suspicion is that it’s a matter of craftsmanship. No one makes plastic clarinets with the kind of care and craftsmanship that they put into a good wooden clarinet. My good clarinet is
built around what they call a polycylindrical bore. What that means is that the body isn’t actually a long cylinder from the mouthpiece to the bell: the exact diameter varies. So you’ve got a very complex shape, and every contour of that shape has an effect. That distinction, the math supports very clearly: change the shape of the body, and you are affecting the waveform of the sound.

Anyway… I finally decided to try one of the M&E plastics. One thing about wooden flutes is that the shape isn’t as complex as a modern boehm clarinet. It’s a conical bore, with very straight lines. So if you made it really carefully, with a really clean, well crafted bore – well, maybe it would work! My plan was to find out about how much it cost, and how long the waitlist was, and then to order one when the next royalty check from my book came in. So I wrote to Michael through email about his polymer flutes. He sells them for just 500 euros, which is astonishingly cheap. (Like I said, the wood ones go for $4000, and most of that cost is the keywork – a custom made keyless costs around $1500; a keyed more than double that.) So I was planning on getting one, if he’d let me return it if I didn’t like it.

And then, he offered to give me one in exchange for building a new website for him. I accepted. So the flute I’m talking about here was given to me by Michael. I didn’t pay for it. But I did not make any promises about what I would say about it.

I’ve had Michael’s flute for a few months now, and… I really can’t believe how good it is. Every time I play it, I’m absolutely stunned by how wonderful it sounds. Over the years, I’ve bought a couple of antique flutes that needed repair… none of them were in good shape – they needed keywork, but they were playable. My M&E has them beat, hands down. It’s not quite up with my Olwell – but it’s amazingly close. Seriously, it comes very close to my Olwell in both sound quality, and sound flexibility. And that’s simply shocking: this flute costs one-half of the cost of a keyless Olwell – and yet, fully keyed, it manages to come close. I’m not going to give up my Olwell for keyless playing, but… if I were starting over and buying a good flute for the first time? I’m not completely sure, but I’d probably go with the keyed M&E.

It’s got excellent sound flexibility. By working with my embouchure, I can easily range from a great reedy sound to a very clear, bright, almost whistle-like sound. It’s very stable in both octaves, and easy to break between. The low D isn’t quite as strong as the low D on the Olwell – it takes a bit of work to get a good hard low D, but it is definitely doable.

For most of the range, the intonation is terrific. All of the standard notes are well-tuned. The only tuning glitch is that the keyed notes on the foot – the low C and C-sharp, are very sharp. But that’s easily fixed – with the foot pulled out on it’s joint just a quarter inch or so, they sound right-on, and it doesn’t seem to effect the low D. Still, that’s a problem. Really, that low key foot should be a quarter inch longer. This really bugs me: in general, everything about this flute is so wonderful, there’s so much care about the aspects of the flute that affect its sound, and yet… the foot is too short. I don’t understand it. You can easily work around it – but it’s frustrating and frankly, kind of sloppy.

It’s very comfortable to play. I’m not sure how he did it, but compared to either an old Rudall and Rose (the style of antique flute I’ve bought) or a new Pratten-style Olwell, the hole size and spacing are very comfortable, without any loss of sound quality. It’s also light. Based on what I knew about polymers before, I was expecting it to be a heavy instrument. It’s heavier than my keyless Olwell, but lighter than my keyed 19th century flutes.

The workmanship is mixed. In terms of things that affect the sound quality, it’s very good. The embouchure hole is cut cleanly, and shaped very well. The fingerholes are clean and well positioned. The keywork is very sturdy and well made, and easy to work with. Pads and springs are all set up properly – the key-springs have the correct tension to keep the pads securely closed while keeping it easy to work the keys quickly. The low foot keys have a roller to make it more comfortable to quickly shift between the low notes in common scale patterns.

Cosmetically, it’s a bit iffy. There are a few scratches around the embouchure hole. Nothing obvious, and certainly nothing that has an effect on its playing. But it’s a tell-tale sign that there’s not quite the same degree of care in making it as you’d find in one of the custom flutes from someone like Pat Olwell.

The joints are strange. Instead of doing something like a wood flute, and putting in a cork ring, he just shaped the polymer into the joints. So the joints are tight, bare polymer. They’re a bit hard to put together, and grease on the joints doesn’t stick particularly well, you’ll get globs of grease getting squeezed out of the joint inside the flute when you put it together. It came with some sort of grease in the joints that’s unpleasant – more like a vaseline than a cork grease. After experimenting a bit, I’ve found that traditional cork grease really doesn’t work well on the plastic – you do need to use something stickier, like a petroleum jelly. This is the one thing about the flute that I really don’t like: the bare polymer joints are, without a doubt, inferior to a corked joint.

The keywork is very nicely done. It’s post-mounted keys. The keys are well made, with good post positioning, good key positioning, springwork set up to make the keys close solidly, without being too tense to open easily. The padwork is excellent. (Which is a bit of a bugaboo of mine. As a long-time clarinetist, I’ve done a lot of pad work, and I’ve found that a lot of people are really sloppy about how they set pads. These are leather-covered pads, set solidly and levelly.)

There are metal rings around the joint edges. The rings around the joints are a bit messy. Again, it’s cosmetic, not functional. But when you look closely around any of the rings, you can see that the polymer isn’t quite flush, and many of the rings have a bit of scratching around them.

The end-cap on the headjoint is ugly. It’s a molded replica of a Rudall&Rose cap, with M&E added on the bottom. Frankly, it’s ugly and cheap looking. Very disappointing, because over all, until you look very closely, the flute is beautiful. There are minor cosmetic problems with the joint rings, but overall, it’s lovely. But that end-cap? It looks terrible. It’s totally unimportant, but for a couple of extra bucks, I’ll bet you could make a much nicer looking endcap.

The material is interesting. The thing that M&E is known for is making pseudo-wood flutes. That is, it’s built in the style of a wooden flute, but they actually use a polymer. It’s black, and it shows the marks of being worked in a way that really looks a lot like wood. Honestly, if I was looking at someone else playing it, I probably wouldn’t guess that it was polymer unless someone told me.

When you pick it up, you know it’s not wood. It doesn’t feel like wood. The main difference is that it feels too smooth – there’s no grain to it. And up close, you can see that the color is too uniform. In real wood, when you look closely, you can always see a bit of color variation. This is just perfectly, uniformly, black. But in terms of weight? It feels like a wooden flute. It’s just a hair heavier than my Olwell – which makes sense, given that it’s carrying full keywork.

It feels rock solid. As an experiment, I tried to scratch the inside of one of the joints with my fingernail. It’s much too hard to scratch like that. It’s a good, solid material. Like most plastics, it’s weatherproof – so you don’t need to worry about humidifying the case, or oiling the wood. And, unlike my Olwell, there’s no variation in playability with the weather. My Olwell sounds different during the winter, due to the dryness of the air. There are noticeable day-to-day variations in how easily certain notes – particularly that all-important strong-low-D – sound. In the M&E, it doesn’t vary: it’s uniformly great.

Of course, the most important thing is the sound. This sounds like a wood flute. It really does. It sounds better than any of the beaten-up real wooden flutes that I’ve acquired. As I said, in terms of sound, it’s not quite up there with my Olwell, but I think that that’s more a matter of workmanship than material. Pat makes a magnificent instrument, and making something not quite as good is absolutely not a critique of M&E.

Being realistic, M&E is selling keyed polymer flutes for 500 euro. Pat made me my keyless wooden flute for something around $1500. For a keyed flute, Pat (and most other makers) charge in the $4,000 range. The M&E is unbelievable when you work price into the equation. It’s better than any of the antiques I’ve played. It’s as good as real wooden keyed flutes by some of the other makers (Sweet and Healy) that I’ve tried. It’s not as good as an Olwell, but for 1/5th the price, and no waiting list? It’s worth every penny it costs and more. It’s a really lovely flute, with a beautiful sound. The workmanship is great where it counts. The cosmetics could use a bit of work – but when you consider the price, that’s really no big deal. Still… if he charged six or seven hundred euros, he’d still be under a fifth the price of a good wooden keyed flute, and he’d be able to fix up some of the cosmetics. I’d definitely be willing to pay an extra one hundred euros for cork joints. (I really hate the uncorked tenons!)

If I had the money, and I could get an Olwell keyed flute tomorrow, I’d probably go for it over the M&E. But given that coming up with money to buy an instrument for my hobby isn’t easy, the huge price difference, the multiyear waiting list? M&E wins. I’m very happy with my M&E. And given a choice between the M&E and pretty much anything but an Olwell? I’d take the M&E happily. I would happily pay Michael for this flute, and I just might end up buying one of his F flutes, to have something with a smaller finger reach.

M&E’s current site has sound samples in realplayer format. I’m working on setting up a new site for M&E. Assuming he approves the design, it should be up by next week. I’ll have updated sound samples in mp3 format, and Michael even sent me a video of Matt Molloy (one of the finest Irish flutists in the world) playing an M&E, which will be on the new site.

Topological Spaces and Continuity

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In the last topology post, I introduced the idea of a metric space, and then used it to define open and closed sets in the space.

Today I’m going to explain what a topological space is, and what continuity means in topology.

A topological space is a set X and a collection T of subsets of X, where the following conditions hold:

  1. \emptyset \in T \land X \in T:both the empty set and the entire set T are in the set of subsets, T. X is going to be the thing that defines the structure of the topological space.
  2. \forall C \in {\bf 2}^T: \bigcup_{c \in C} \in T: the union of collection of subsets of T is also a member of T.
  3. \forall s,t \in T: s \cap t \in T: the intersection of any two elements of T is also a member of T.

The collection T is called a topology on X. The members of T are the open sets of the topology. The closed sets are the set complements of the members of T. Finally, the elements of the topological space X are called points.

The connection to metric spaces should be pretty obvious. The way we built up open and closed sets over a metric space can be used to produce topologies. The properties we worked out for the open and closed sets are exactly the properties that are required of the open and closed sets of the topology.

The idea of the topology X is that it defines the structure of X. We say collection when we talk about it, because it’s not a proper set: a topology can be (and frequently is) considerably larger than what’s allowable for a set.

What it does is define the notion of nearness for the points of a set. Take three points in the set X: a, b, and c. X contains a series of open sets around each of a, b, and c. At least conceptually, there’s a smallest open set containing each of them. Given the smallest open set around a, there is a larger open set around it, and a larger open set around it. On and on, ever larger. Closeness in a topological space gets its meaning from those open sets. Take that set of increasingly large open sets around a. If you get to an open set around a that contains b before you get to one that contains c, then b is closer to a than c is.

There are many ways to build a topology other than starting with a metric space, but that’s definitely the easiest way. One of the most important ideas in topology is the notion of continuity. In some sense, it’s the fundamental abstraction of topology. Now that we know what a topological space is, we can define what continuity means.

A function from topological space T to topological space U is continuous if and only if for every open set C subseteq U, the inverse image of f on C is an open set.

Of course that makes no sense unless you know what the heck an inverse image is. If C is a set of points, then the image f(C) is the set of points { f(x) | x \in C }. The inverse image of f on C is the set of points { x | f(x) \in C}.

Even with the definition, it’s a bit hard to visualize what that really means. But basically, if you’ve got an open set in U, what this says is that anything that maps to that open set must also have been an open set. You can’t get an open set in U using a continuous function from T unless what you started with was an open set. What that’s really capturing is that there are no gaps in the function. If there were a gap, then the open spaces would no longer be open.

Think of the metric spaces idea of open sets. Imagine an open set with a cube cut out of the middle. It’s definitely not continuous. If you took a function on that open set, and its inverse image was the set with the cube cut out, then the function is not smoothly mapping from the open set to the other topological space. It’s mapping part of the open set, leaving a big ugly gap.

If you read my old posts on category theory, here’s something nifty.

The set of of topological spaces and continuous functions form a category, with the spaces as objects and continuous functions as arrows. We call this category Top.

Aside from the interesting abstract connection, when you look at algebraic topology, it’s often easiest to talk about topological spaces using the constructs of category theory.

For example, one of the most fundamental ideas in topology is homeomorphism: a homeomorphism is a bicontinuous bijection (a bicontinuous function is a continuous function with a continuous inverse; a bijection is a bidirectional total function between sets.)

In terms of the category {\bf Top}, a homeomorphism between topological spaces is a homomorphism between objects in Top. That much alone is pretty nice: if you’ve gotten the basics of category theory, it’s a whole lot easier to understand that a homeomorphism is an homo-arrow in {\bf Top}.

But there’s more: from the perspective of topology, any two topological spaces with a homeomorphism between them are identical. And – if you go and look at the category-theoretic definition of equality? It’s exactly the same: so if you know category theory, you get to understand topological equality for free!

Electoral Rubbish

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And now, for your entertainment, a bad math quickie.

I live in New York. ’round here, we’ve got a somewhat peculiar feature of how we run our elections. A single candidate can run for office on behalf of multiple parties. If they do, they appear on the ballot in multiple places – one ballot line for each party that they represent. When votes are tallied, if the candidate names for two different ballot lines match exactly, then the votes for those two lines are combined.

The theory behind this is that it allows people to say a bit more with their votes. If you want to vote for the democratic candidate, but you also want to express you preferences for policies more liberal than those of the democratic party platform, you can vote for the democrat, but do it on the liberal party line instead of the democratic party line.

In practice, what this means is that we’ve got lots of patronage parties – that is, lots of small parties which were set up by a small group of people as a way of making money by, essentially, selling their ballot line.

One thing we hear, election after election, is how terribly important these phony parties are. This year, we keep on hearing, over and over, how no Republican has won a statewide election since 1975 without the backing of the Conservative party! Therefore, winning the backing of the Conservative party is so very, very important!

This is, alas, a classic example of the old problem: correlation does not imply causation. The Republicans don’t lose elections because they don’t have the backing of the Conservative party: the Conservative party always backs the republican candidate unless it’s completely clear that they’re going to lose.

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Return of a Classic: The Electromagnetic Gravity Revolution!

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Between work, trying to finish my AppEngine book, and doing all of the technical work getting Scientopia running smoothly on the new hosting service, I haven’t had a lot of time for writing new blog posts. So, once again, I’m recycling some old stuff.

It’s that time again – yes, we have yet another wacko reinvention of physics that pretends to have math on its side. This time, it’s “The Electro-Magnetic Radiation Pressure Gravity Theory”, by “Engineer Xavier Borg”. (Yes, he signs all of his papers that way – it’s always with the title “Engineer”.) This one is as wacky as Neal Adams and his PMPs, except that the author seems to be less clueless.

At first I wondered if this were a hoax – I mean, “Engineer Borg”? It seems like a deliberately goofy name for someone with a crackpot theory of physics… But on reading through his web-pages, the quantity and depth of his writing has me leaning towards believing that this stuff is legit. (And as several commenters pointed out the first time I posted this, in Germany, you need a special license to be an engineer, and as a result, “Engineer” is actually really used as a title. Still seems pompous to me – I mean, technically, I’m entitled to go around calling myself Dr. Mark Chu-Carroll, PhD., but I don’t generally do that.)

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The Return of a Classic: Neal Adams' Bad Physics

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Between work, trying to finish my AppEngine book, and doing all of the technical work getting Scientopia running smoothly on the new hosting service, I haven’t had a lot of time for writing new blog posts.

But in the process of doing my technical work around here, I was browsing through some archives, and seeing some of my old posts that I’d forgotten about. And odds are, if I forgot about it, then there are a lot of readers who’ve never seen it. So I’m going to bring back some of the classic old material.

For example, Neal Adams. Comic book fans will know about Neal: he’s a comic book artist who worked on some of the most famous comics in the 1970s: he drew Batman, Superman, Deadman, Green Lantern, the Spectre, the X-men. More recently, he’s done a lot of work in general commercial art – for example, he did the animated nasonex bee commercials a few years ago.

Adams' PMP image But he’s not just an artist. No, he’s so much more than that! He’s also a brilliant scientist. He’s much smarter than all of those eggheads with college degrees. They’re struggling to build giant particle accelerators to help understand things like mass. But Neal – he’s got them beat. He’s figured out exactly how things work!

According to Neal, there is no such thing as gravity – it’s all just pressure. People trying to figure out stuff about how gravity works are just wasting time. The earth (and all other planets) is actually a matter factory – matter is constantly created in the hollow center of the earth, and the pressure of all the new matter forces the earth to constantly expand. The constant expansion creates pressure on the surface as things expand – and that constant expansion is what creates gravity! You’re standing on a point on the surface of the earth. And the earth is expanding – the ground is pushing up on you because of that expansion. You’re not being pulled down towards the earth: the earth is pushing up on you.

And according to Neal, the best part is the math works!. In the original version of this post, I had a link to Neal’s page with his explanation of how the math works – but he has, since then, moved most of his science stuff behind a paywall – you now need to pay Neal $20 to get to see his material, so I can’t provide a direct link. But it’s in a video here, and you can see the original using the Wayback Machine.

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The Halting Problem

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Some people expressed interest in seeing a full-out, formal presentation of the Halting problem proof. So, I’m going to walk through it. There are actually a lot of different versions of this proof; the one that I’m going to use is based on the one used by my grad-school theory professor, Dr. John Case. To be clear, I’m doing it from memory, so any errors are completely my own fault, not John’s!

To start off, we need to define what a computing device is. In formal mathematical terms, we don’t care how it works – all we care about is what it can do in abstract tems. So what we do is define something called an effective computing device. An effective computing device is any Turing equivalent computing device. An ECS is modelled as a two parameter function phi : {cal N} times {cal N} rightarrow {cal N}. The first parameter is an encoding of a program as a natural number; the second parameter is the input to the program. It’s also a natural number, which might seem limiting – but we can encode any finite data structure as an integer, so it’s really not a problem. The return value is the result of the program, if the program halts. If it doesn’t halt, then we say that the pair of program and input aren’t in the domain of phi. So if you wanted to describe running the program “f” on the input 7, you’d write that as phi(f, 7). And, finally, the way that we would write that a program p doesn’t halt for input i as phi(p, i) = bot.

So now we’ve got our basic effective computing device. There’s one thing we still need before we can formulate the halting problem. We need to be able to deal with more parameters. After all – a halting oracle is a program that takes two inputs – another program, snd the input to that program. the easiest way to do that is to use something called a pairing function. A pairing functions is a one-to-one function from an ordered pair to an integer. There are lots of possible pairing functions – for example, you can convert both numbers to binary, left-pad the smaller until they’re equal length, and then interleave the bits. Given (9,3), you convert 9 to 1001, and 3 to 11; then you pad 3 to 0011, and interleave them to give you 10001011 – 139. We’ll write our pairing function as angle brackets around the two values: langle x,yrangle.

With the help of the pairing function, we can now express the halting problem. The question is, does there exist a program O, called a halting oracle, such that:

forall p, forall i: (varphi(O,langle p,irangle) = left{ begin{array}{l} 0 mbox{ if } varphi(p, i) = bot \ 1 mbox{ if } varphi(p,i) neq bot end{array}right.

In english, does there exist a program O such that for all pairs of programs p and inputs i, the oracle returns 1 if varphi(p, i) halts, and 0 if it doesn’t?

Time, finally, for the proof. Suppose that we do have a halting oracle, O. That means that for any program p and input i, varphi(O, langle p, i rangle) = 0 iff varphi(p,i)=bot.

So, can we devise a program $p_d$ and input i where varphi(p_d, i) != bot,
but varphi(O, langle p, i rangle) = 0?

Of course we can. We need a p_d which takes two parameters: an oracle, and an input. So it should be really simple right? Well, not quite as easy as it might seem. You see, the problem is, p_d needs to be able to pass itself to the oracle. But how can it do that? A program can’t pass itself into the oracle. Why not? Because we’re working with the program as a Gödel number – and a program can’t contain its own Gödel number. If it contained it, it would be larger than itself. And that’s rather a problem.

But there’s a nice workaround. What we care about is: is there any combination of program and input such that O will incorrectly predict the halting status? So what we’ll do is just turn p_d into a parameter to itself. That is, we’ll look at a program p_d like the following:

def deceiver(input):
   (oracle, program) =  unpair(input)
   if oracle(program, input):
      while(True): continue
   else:
      halt

Then we’ll be interested in the case where the value of the program parameter is a Gödel number of the deceiver itself.

(As an aside, there are a variety of tricks to work around this. One of the more classical ones is based on the fact that for any given program, p, there are an infinite number of versions of the same program with different Gödel numbers. Using that property, you can embed a program p_{d_2} into another program p_{d}. But there are a few other tricks involved in getting it right. It’s not simple – Alan Turing screwed it up in the first published version of the proof!)

Now, when input = langle O, p_d rangle, then O will make the wrong prediction about what p_d will do. So – once again, we’re back where we were in the simpler version of the proof. A halting oracle is a program which, given any pair of program and input, will correctly determine whether that program will halt on that input. We’re able to construct a pair of program and input where the oracle doesn’t make the right prediction, and therefore, it isn’t a halting oracle.

This version is more abstract, but it’s still got that wonderfully concrete property. Even in the most abstract way of talking about a computing device, if you’ve got something that you believe is a halting oracle, this shows you how to create a program that will prove that the halting oracle is wrong. So you can’t possibly create a halting oracle.

And to be extra clear: this doesn’t rely on any ambiguities about definitions, or any distinction between values and meanings. It shows a way of producing a real, concrete counterexample for any purported halting oracle. No trickery, no fudging – if you think you have a halting oracle, you’re wrong, and this proof shows you exactly how to create a program that will demonstrate that.

Turing Crackpottery!

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One of the long-time cranks who’s commented on this blog is a bozo who goes by the name “Vorlath”. Vorlath is a hard-core Cantor crank, and so he usually shows up to rant whenever the subject of Cantor comes up. But last week, while I was busy dealing with the Scientopia site hosting trouble, a reader sent me a link to a piece Vorlath wrote about the Halting problem. Apparently, he doesn’t like that proof either.

Personally, the proof that the halting problem is unsolvable is one of my all-time favorite mathematical proofs. It’s incredibly simple – just a couple of steps. It’s very concrete – the key to the proof is a program that you can actually write, easily, in a couple of lines of code in a scripting language. And best of all, it’s incredibly profound – it proves something very similar to Gödel’s incompleteness theorem. It’s wonderful.

To show you how simple it is, I’m going to walk you through it – in all of its technical details.

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