Tuesday, March 31, 2015

Mathematical Cranks

(Crossposted from my Facebook page)

I'm reading a book called "Mathematical Cranks". It's an anthology summarizing a sample of different people, ranging from the mildly eccentric to the downright insane, who have tried to put forward their bad ideas as legitimate mathematical research.

A couple examples: A guy who claims that the number 57 is intimately linked with every major event in US history; numerous people claiming to have found fault in existing theorems; and a bunch of people who claim to have a Grand Unified Theory of Mathematics, which will change the world and will solve every problem in existence, provided that you purchase their book for 29.99. A modern example would be the Time Cube guy, or roughly half the people listed in the Wikipedia article about Usenet celebrities.

I'm about 56 pages in, and so far, all the cranks mentioned in this book have been male. Further evidence that we need to do more to remove the stigma around women entering the field of fake mathematics.

I get the sense that the author is being as generous as possible to the people he talks about, which I like -- I was worried the book would just be a series of "Hey, let's all point and laugh at this idiot for being bad at math!" but that hasn't been the case.

For example, there's a chapter on the Dozenal Society, a group of people who want everyone to switch to a Base 12 number system instead of Base 10, with the reasoning that Base 12 makes a lot of calculations easier. The author admits that these people aren't *wrong*, per se, but it's still never gonna happen. There are other cases where the crank in question has found something mildly interesting, but it's in the middle of a series of grandiose claims.

Part of the reason I was interested in reading this book is because I'm always interested in figuring out ways I hadn't considered in which my thinking could be wrong. In my last Facebook post, I mentioned that my high school math teacher made a joke about how, if I didn't keep my thinking under control, I would end up like Howard Hughes. I'd like to avoid that fate, if I can (the part where his mental illness drove him to become a paranoid recluse, not the part about being a genius billionaire playboy philanthropist).

And of course, there are a lot of mathematicians who end up losing their sanity obsessing over something or other. The best example that comes to mind is Georg Cantor, who spent his later years alternating between staying in mental hospitals and obsessing over the theory that Francis Bacon is the actual author of all the Shakespeare plays. But the subject of mathematicians and mental illness is a topic for another day.

But a recurring theme I've noticed among the cranks is that the most grandiose claims seem to come from the people who do the least actual mathematics. They spend most of their time talking about how great and powerful their theory is, how it makes all other mathematics obsolete, but then their "theory" turns out to be a couple of equations that aren't really that impressive, without elaborating on how to use these findings.

In other cases, the crank will make a grandiose claim, but fail to actually prove it formally. This is kind of a no-no in mathematics -- not that every finding always requires a formal proof, but if you don't have a proof, at least have the decency to call your finding a "conjecture" rather than a "Grand Unifying Theory that will Change the World". I can't find the quote, but I remember a mathematics blogger saying "The worst kind of mathematics is no mathematics at all" and that seems to be the case here.

Tuesday, March 24, 2015

Phil Robertson, Conformity and You

Phil Robertson (don't stop reading yet, I swear this will be interesting) said something stupid recently. It is something that I don't think anybody I know actually agrees with or takes seriously. There might be a few contrarians among my friends who are interested in taking Phil Robertson's views to their logical conclusion as a purely academic exercise (I'm a bit of a contrarian myself sometimes), but that's not the same thing as sincerely believing it.

And yet, clearly there are people who take Phil Robertson seriously. I guarantee you that swarms of people will jump to his defense, for everything stupid he's said in the past or will say in the future. I'm not talking about defending freedom of speech for views you hate (I don't think he should be arrested, I just think it's sad that he's probably going to live the rest of his life and die as a hate-filled idiot -- an entire human life wasted). I'm talking about people who actively defend his views and believe that people should agree with him.

So, my first question is, why have I never met one? With Duck Dynasty being as big an empire as it is, I would have to think that, even in a place like BC, there must be a non-trivial percentage of people who totally agree with Phil Robertson's worldview. Statistically, I must have walked past a few of them in the street at some point. These people clearly exist. I've just never had the pleasure of having a conversation with one.

I have about 300 Facebook friends. As far as I know, none of these people share Phil Robertson's worldview.

I don't know what percentage of the population of BC shares something similar to Phil Robertson's worldview -- archetypical, old-fashioned, bigoted, religious extremism. Let's say BC is 1% "Robertsonians" to make up a word (I'm just making up a number here, but as they say, if something's worth doing, it's worth doing with fake statistics. I doubt it's as high as 10% or as low as 0.1%).

Then if my friends were selected for me randomly, the odds that a randomly chosen friend is a Robertsonian is 0.01, which means that the odds they are not a Robertsonian is 0.99. Then the odds that all 300 of my friends are not Robertsonians is 0.99^300, which is close to 0.05. Not impossible, but still pretty unlikely, IF we assume that it occurred randomly.

Obviously, it didn't occur randomly. One huge factor is that I've spent the last 6 years in post-secondary education, so I've been spending my time around educated people. But it would be dishonest to ignore the possibility that, like it or not, I'm most likely to find myself gravitating towards people who, for the most part, share my values. If nothing else, I typically don't actively seek out people whose values clash with mine (look up The Parable of the Polygons to see the problem with that).

So: Consciously or unconsciously, I have excluded people from my life because they don't share my values.

On the one hand, I don't expect any of you will think less of me for not wanting to be buddies with Phil Robertson. On the other hand, when you phrase it like that ("excluding people from my life because they don't share my values") I sound like a bit of a jerk. Isn't it supposed to be a good thing to spend time with people who have different values than you have so you can be open-minded and learn from them?

In spite of people like Phil Robertson, I think the answer *might* be yes. It's just not easy to do. You know that anger you feel when someone like Phil Robertson comes along and says something you think is bigoted and ignorant? That's how it feels when you come across a person whose values ACTUALLY clash with yours.

Remember my last long post, where I made fun of people who say "I'm open-minded, but only to good ideas"? Its close cousin is "I'm tolerant of other people's values, unless I think they're going too far" -- if that's all it takes to be tolerant, then Phil Robertson is tolerant too. In fact, so is pretty much every person in the world.

You might wonder why I'm so obsessed with being open-minded to stupid ideas. I'm obsessed with it because it would be so easy to imagine a world where the roles are switched: 99% of the people in BC are the clean, decent, God-fearing Phil Roberts followers, and the other 1% are the frightening, militant, liberal secular humanists who everyone hates. In such a society, would you stick with what you know is right, or would you find yourself convinced by the Robertsonians?

Bear in mind, they have all the cool and powerful people, not to mention all their good columnists are writing thinkpieces about how secular humanists are Problematic. Plus, there are people on the internet who, upon finding a tweet you made expressing your heathen views, will do their best to report it to your employer so you lose your job. What if the people who agree with you are only 0.1% of the population, or 0.001%? What if it was just you?

And -- here's the Twilight Zone twist -- whatever mode of thinking you would apply in this alternate universe, are you applying the same mode of thinking in this universe?

(Take a moment to look up the Asch experiments on conformity if you haven't already heard of them.)

When I see someone like Phil Robertson, I don't think "Hah, what an idiot" -- not because I don't think he's an idiot, but because it's not very productive. My questions are more along the lines of "How does a human brain come to have those beliefs? Heck, how did I come to believe the things I believe? And, in some hypothetical universe where I was wrong and the Phil Robertson and Fred Phelps' of the world are right, what evidence would it take to convince me of it?"

If I have a point here, it's that Phil Robertson's brain is not fundamentally different from yours or mine, and that should terrify you, or at least motivate you to consider how much of your own thinking is in error.

Friday, March 6, 2015


I occasionally hear someone say "I'm open-minded, but only to good ideas." This implies that, as soon as they're exposed to an idea, they *immediately* know whether it's a good idea or not even without taking any time to consider it.

This is an amazing and miraculous talent to have -- the ability to instantly recognize an idea as good or bad. I, for one, usually have to *think* for at least a couple seconds before I know with any certainty whether an idea is good or not (hell, even when I *know* an idea is bad, I usually try to think out its consequences anyway just to make sure I'm not missing anything).

So, the next time I meet someone who says that, I'm going to tie them up in front of a computer screen and force them to watch a program that generates every series of English words in increasing lexicographic order. By their own claim, whenever the series of words describes a good idea, their response will be open-mindedness, which has a somatic feeling which I suspect would lend itself to being detected by an existing scan, possibly fMRI.

The principle is this: you show this person a sentence describing an idea, e.g. "Increase funding of space exploration by X dollars". If the person is open-minded to it, then it's a good idea. If not, reject it. Move on to next sentence.

I know this sounds extreme, but consider all the potential good ideas that nobody has detected yet. Cold fusion, feasible quantum computing, cures for all known diseases -- it would be an extreme act of stupidity NOT to at least try it. Think of the billions of lives that could be saved or improved, compared to the miniscule cost of this one person's freedom and wellbeing.

The human visual system can quite comfortably process visual information at 10 "frames per second" for lack of a better term. Provided we feed the person while they're staring at the screen, and allow them to sleep about a third of the time, I figure we can have them looking at ideas for about 16 hours a day, and we could probably keep the person alive for about 50 years if we had to (and we DO have to -- we can't let an opportunity like this go to waste).

So, at 10 ideas per second, 16 hours a day for 50 years, that adds up to them being exposed to about 10 billion ideas in a lifetime. One issue is the sheer number of possible English sentences. It is estimated that each letter in English represents about 0.6 to 1.2 bits of information (I'll round it to 1 bit). So, with a little cleverness, we would be guaranteed to expose this person to every 33-character English sentence if we wanted to. If we restrict it even more to get rid of obviously nonsensical phrases or restrict ourselves to vocabulary in the fields of ideas we want evaluated (e.g. medicine, engineering, economics) we could probably get even more value out of them.

I mean, unless they're lying about being open-minded, or about being open-minded only to good ideas. But I'm pretty sure nobody would do that.
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It Seemed Funny at the Time by Ben Buckley is licensed under a Creative Commons Attribution-Noncommercial 2.5 Canada License.