I assume that Comic Strip Master Command ordered many mathematically-themed comic strips to coincide with the United States school system getting back up to full. That or they knew I’d have a busy week. This is only the first part of comic strips that have appeared since Tuesday.
Mel Henze’s Gentle Creatures for the 7th and the 8th of September use mathematical talk to fill out the technobabble. It’s a cute enough notion. These particular strips ran last year, and I talked about them then. The talk of a “Lagrangian model” interests me. It name-checks a real and important and interesting scientist who’s not Einstein or Stephen Hawking. But I’m still not aware of any “Lagrangian model” that would be relevant to starship operations.
Jon Rosenberg’s Scenes from a Multiverse for the 7th of September speaks of a society of “powerful thaumaturgic diagrammers” who used Venn diagrams not wisely but too well. The diagrammers got into trouble when one made “a Venn diagram that showed the intersection of all the Venns and all the diagrams”. I imagine this not to be a rigorous description of what happened. But Venn diagrams match up well with many logic problems. And self-referential logic, logic statements that describe their own truth or falsity, is often problematic. So I would accept a story in which Venn diagrams about Venn diagrams leads to trouble. The motif of tying logic and mathematics into magic is an old one. I understand it. A clever mathematical argument often feels like magic, especially the surprising ones. To me, the magical theorems are those that prove a set of seemingly irrelevant lemmas. Then, with that stock in hand, the theorem goes on to the main point in a few wondrous lines. If you can do that, why not transmute lead, or accidentally retcon a society out of existence?
Mark Anderson’s Andertoons for the 8th of September just delights me. Occasionally I feel a bit like Mark Anderson’s volunteer publicity department. A panel like this, though, makes me feel that he deserves it.
Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 8th of September is the first anthropomorphic-geometric-figures joke we’ve had here in a while.
Mike Baldwin’s Cornered for the 9th of September is a drug testing joke, and a gambling joke. Both are subjects driven by probabilities. Any truly interesting system is always changing. If we want to know whether something affects the system we have to know whether we can make a change that’s bigger than the system does on its own. And this gives us drug-testing and other statistical inference tests. If we apply a drug, or some treatment, or whatever, how does the system change? Does it change enough, consistently, that it’s not plausible that the change just happened by chance? Or by some other influence?
You might have noticed a controversy going around psychology journals. A fair number of experiments were re-run, by new experimenters following the original protocols as closely as possible. Quite a few of the reported results didn’t happen again, or happened in a weaker way. That’s produced some handwringing. No one thinks deliberate experimental fraud is that widespread in the field. There may be accidental fraud, people choosing data or analyses that heighten the effect they want to prove, or that pick out any effect. However, it may also simply be chance again. Psychology experiments tend to have a lower threshold of “this is sufficiently improbable that it indicates something is happening” than, say, physics has. Psychology has a harder time getting the raw data. A supercollider has enormous startup costs, but you can run the thing for as long as you like. And every electron is the same thing. A test of how sleep deprivation affects driving skills? That’s hard. No two sleepers or drivers are quite alike, even at different times of the day. There’s not an obvious cure. Independent replication of previously done experiments helps. That’s work that isn’t exciting — necessary as it is, it’s also repeating what others did — and it’s harder to get people to do it, or pay for it. But in the meantime it’s harder to be sure what interesting results to trust.
Ruben Bolling’s Super-Fun-Pak Comix for the 9th of September is another Chaos Butterfly installment. I don’t want to get folks too excited for posts I technically haven’t written yet, but there is more Chaos Butterfly soon.
Rick Stromoski’s Soup To Nutz for the 10th of September has Royboy guess the odds of winning a lottery are 50-50. Silly, yes, but only because we know that anyone is much more likely to lose a lottery than to win it. But then how do we know that?
Since the rules of a lottery are laid out clearly we can reason about the probability of winning. We can calculate the number of possible outcomes of the game, and how many of them count as winning. Suppose each of those possible outcomes are equally likely. Then the probability of winning is the number of winning outcomes divided by the number of probable outcomes. Quite easy.
— Of course, that’s exactly what Royboy did. There’s two possible outcomes, winning or losing. Lacking reason to think they aren’t equally likely he concluded a win and a loss were just as probable.
We have to be careful what we mean by “an outcome”. What we probably mean for a drawn-numbers lottery is the number of ways the lottery numbers can be drawn. For a scratch-off card we mean the number of tickets that can be printed. But we’re still stuck with this idea of “equally likely” outcomes. I suspect we know what we mean by this, but trying to say what that is clearly, and without question-begging, is hard. And even this works only because we know the rules by which the lottery operates. Or we can look them up. If we didn’t know the details of the lottery’s workings, past the assumption that it has consistently followed rules, what could we do?
Well, that’s what we have probability classes for, and particularly the field of Bayesian probability. This field tries to estimate the probabilities of things based on what actually happens. Suppose Royboy played the lottery fifty times and lost every time. That would smash the idea that his chances were 50-50, although that would not yet tell him what the chances really are.