Utility: an introduction for ivory tower mathematicians

I identify as utilitarian—

—which is not to say that I believe in the weird sort of aggregative add the happiness scores moral codes that prefer a society of a billion slaves and one obscenely happy person to today's world, but which is to say that I believe (approximately) that the highly complicated human notion(s) of good obey(s) the von Neumann–Morgenstern axioms given sufficient computational resources and bias correction—

—and so I decided I may as well get around to explaining and playing around with the VNM axioms and a couple of little mathematical-philosophical thoughts surrounding them. And yes, this is hardly groundbreaking stuff never before seen on the Internet, but hey, it could be worse: I could be trying to explain what Haskell monads are.

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What things have utility?

Let \(\mathcal{O}\) be the set of outcomes. These represent events like the cat lives and the cat dies, as well as more probabilistic events like 50% chance the cat lives.

[Review] The House of Silk, by Anthony Horowitz

It was almost dark and with the coming of the night the sense of ease that I had felt had quite dissipated, and the city had once again turned cold and hostile. The shoppers and the entertainers had all gone home and their places had been taken by a different species altogether, shabby men and gaudy women who needed shadows in which to conduct their business and whose business, in truth, carried shadows of its own.

The House of Silk, by Anthony Horowitz (2011)

Cover: Orion, 2012 edition

Anthony Horowitz's The House of Silk is a contemporary addition to Sherlock Holmes canon that is in every way in the spirit of Arthur Conan Doyle's original stories, which, despite having had a strong influence on English-language crime fiction, have a distinctive style of their own which doesn't quite fit in the same mold as most mystery tales. It's not a six-character locked door whodunnit in the vein of Christie or Sayers, nor does it have the bombastic cartoonishness of the Robert Downey Jr. movies; instead, The House of Silk is a thing of its own: a crime adventure filled with twists, intrigue and a level of scandal that would titillate any Victorian reader (not that that'd be much of a feat).

Man logic vs. woman logic

Notes on set cardinality

Credit: deviantART / Sc0t0ma

I'm working through LMU's Introduction to Mathematical Philosophy on Coursera at the moment. I stumbled across something that confused me (specifically, the claim that has fewer elements than defines a total order), so I'm using this blog post as a medium in which to work through it methodically enough to explain it to someone else.

(Warning: maths below the fold.)

If you asked your MP one question...

TL;DR: meta-politics.

Credit: Sky News

There's an election on in Australia soon, and if you're 18 or over you may be wondering who to vote for. (Having paid attention in school, you're of course aware that under the Westminster system, you don't vote for a prime minister: you vote for your local MP.)

There's a fairly useful question worth asking your local candidates, and it happens to be perfectly generic:

If push came to shove, what issues would you put the people's values ahead of yours for?

Race to the centre

Seth is running against Olivia for Class President. It's not as exciting as it sounds, because the Class President only really gets to decide one thing: how long lunch time is.

It's a pretty divisive issue. Some of their classmates enjoy playing outside and would love for lunch to be longer. Others are total bookworms and want shorter lunch times so that they can get back to reading The Magic School Bus.

Still a better love story...

You might even say that the whole situation is a bit like a political spectrum:

I had some trouble deciding which way around to make the diagram. On the one hand, longer lunch times are definitely right-wing, because they represent greater social autonomy. On the other hand, longer lunch times are definitely left-wing, because they represent greater social autonomy.

Seth and Olivia have their own personal opinions on the idea lunch time length, which we can mark out on the spectrum like so:

When the election occurs, students will vote for whichever candidate's position is closest to theirs. This means that the outcome of the election is actually very predictable: whoever's policy is closer to the median is going to win.

[Reblog] "Time" (xkcd)

Original post: Creator of xkcd Reveals Secret Backstory of His Epic 3,099-Panel Comic, Wired.

I'm endlessly fascinated by the new possibilities that the Internet opens in storytelling. The coherent crowd-sourced mythology of SCP. The memetic hypertextuality of Homestuck.

Still from Time, xkcd. Retrieved from Wired.

I wasn't really following Randall Munroe's Time, but the nature in which the story was told is fascinating in much the same way. Its 'serial' nature invites, maybe even necessitates, frame-by-frame analysis.

And of course, on top of that, there's the attention to detail and stylistic elegance of Munroe's best xkcd work.

The mathematics of war and dodgeball

Lanchester's Laws are a handful of mathematical formulae describing armed combat between two forces. I'll talk about their derivation for a little bit, then move on to dodgeball (duh).

Ranged combat

(Age of Empires III, Ensemble Studios, 2005)

Two armies of foes face one another. One is armed with rifles; the other with crossbows. Which army prevails?

Let's call the size of the respective armies \(x\) and \(y\), and call their soldiers' lethality \(\alpha\) and \(\beta\) respectively. (The lethality of a soldier is the number of enemy soldiers they can strike down per unit time.)