Monthly Archives: November 2011

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.