Stumbled across 24th July 2013

 2013-07-24

The Orbit/Stabiliser Theorem

The Orbit/Stabiliser Theorem is a simple theorem in group theory. Thanks to Tim Gowers for the proof I outline here - I find it much more intuitive than the proof that was presented in lectures, and it involves equivalence relations (which I think are wonderful things).

Theorem: \(\vert {g(x), g \in G} \vert \times \vert {g \in G: g(x) = x} \vert = \vert G \vert\).

Proof: We fix an element \(x \in G\), and define two equivalence relations: \(g \sim h\) iff \(g(x) = h(x)\), and \(g \cdot h\) if \(h^{-1} g \in \text{Stab}_G(x)\), where \(\text{Stab}_G(k) = {g \in G: g(k) = k}\).

 2013-07-22

On Shakespeare

I’ve now seen two Shakespeare plays at the Globe - once in person, to see A Midsummer Night’s Dream, and once with a one-year-and-eighty-mile gap between viewing and performance (through the Globe On Screen project), to see Twelfth Night. Both times the plays were excellent. Both were comedies, and both were laugh-out-loud funny.

The performance of Twelfth Night, then, was beamed into a local-ish cinema for our viewing pleasure. (Definitely more comfortable than the seating at the Globe, although I am reliably informed that if you go to the Globe, you really have to be a groundling, standing at the front next to the stage, in order to get the proper experience.) My seat was next to those of some young-ish children. The result of taking several young children to a three-hour performance of a play which isn’t in Modern English was predictable, but it got me thinking. (Bear with me - this will become relevant.)

 2013-07-21

My objection to the One Logical Leap view

A large chunk of the reason why changing someone’s mind is so difficult is the fact that our deeply-held beliefs seem so obviously true to us, and we find it hard to understand why those beliefs aren’t obvious to others. Example:

A: A god exists - look around you; everything you see is so obviously created, not stumbled upon! B: No, that’s rubbish - look around you, everything you see is easily explained by understood processes!

 2013-07-18

Prerequisites for hypothetical situations

Usually when I discover (or, more rarely, think up) a thought experiment about a moral point, and discuss it with an arbitrary person whom I will (for convenience) call Kim, the conversation usually goes like this:

Me: {Interesting scenario} - what do you think?

Kim: I would just {avoids point of scenario by nitpicking}

Me: You know what I meant. {applies easy fix to scenario to prevent nitpick}

Kim: Well then, I’d {avoids point of scenario by raising unrelated moral issue}

 2013-07-14

The Multiple Drafts view of consciousness

I’ve been reading one of Daniel Dennett’s books, Consciousness Explained. Aside from the fact that the author has an incredible beard and is therefore correct on all matters, he can also write a very cogent book. In Consciousness Explained, Dennett outlines what he calls the Multiple Drafts approach to explaining consciousness; this blog post is my attempt to summarise that view in a couple of short analogies.

Dennett starts off by providing evidence that our time-perception is somewhat malleable: we can interpret two dots of different colours (appearing separated by a short distance in time and space) as a single moving dot that changes colour abruptly at some point. The key puzzle here is that we perceive the colour to have changed before seeing the second coloured dot. Dennett then outlines what seem to be the two mainstream points of view on how this happens.

 2013-07-14

Stumbled across 13th July 2013

 2013-07-13

A framework for discussing "pricelessness"

Sometimes some people argue that certain things are “priceless” - that is, worth an infinite amount of money to them. I posit that what this really means is that it would take work and uncomfortable imagination to evaluate the worth of that thing to them.

The example that triggered this framework was my evaluation of how much my sense of smell was worth to me. (It was late at night and I couldn’t get to sleep, so I just let my mind wander around for a bit.) I was unable to quantify the amount I would pay to keep my sense of smell, but it is certainly finite, as the following thought experiment demonstrates.

 2013-07-12

Imre Leader Appreciation Society

There was once a small website devoted to noting the more interesting quotes from our more idiosyncratic lecturers. It sadly vanished from the web, although after some detective work, I found a copy floating around on one of Amazon’s servers. I stored them for posterity using the archival service WebCitation, which is itself now dead, so instead I shall link to Konrad Dąbrowski’s capture.

 2013-07-10

Stumbled across 9th July 2013

Being bored over the summer holiday, I decided that I would document the cool things I ran across on the Internet. Over the last week, there have been many of these. If I see anything particularly amazing, it’ll go in one of these aggregation posts.

 2013-07-09

An obvious improvement to tennis

So yesterday the Wimbledon tennis tournament was decided. The system for verifying whether the tennis ball is out or not (and hence whether play for the point stops or continues) on the main courts is as follows:

  1. The ball lands.
  2. The linesperson keeping charge of the line nearest to the landing point of the ball works out whether the ball landed inside or outside the region demarcated by the line.
  3. The umpire decides whether or not to overrule the linesperson’s decision.
  4. The Hawkeye ball-tracking system determines whether the ball landed inside or outside the region demarcated by the line.
  5. If either player disagrees with the official decision (that is, if the linesperson called “out” when the player thought the ball was in, or the linesperson was silent when the player thought the ball was out, or if the umpire overruled a decision that the player thinks was correct) then that player informs the umpire that ey wishes to “challenge” the linesperson. In this instance, the Hawkeye reading is consulted (and the ball’s trajectory slowly animated on a big screen, for added tension) and regarded as definitive.

The problem I have with this system is the process of “challenging”. Each player starts out with a challenge count of three. If a player makes a challenge, and Hawkeye contradicts the official call, then the challenge count is maintained at its current level. If a player makes a challenge, and Hawkeye agrees with the official call, then the challenge count for that player is decremented. A player cannot challenge if eir challenge count is 0. On entering a tie-break, each player’s challenge count is incremented.

 2013-07-08

Mundane magics

I have stumbled across a LessWrong post on the importance of seeing what is real for just how cool it is. It lists such examples as:

  • Vibratory Telepathy.  By transmitting invisible vibrations through the very air itself, two users of this ability can share thoughts.  As a result, Vibratory Telepaths can form emotional bonds much deeper than those possible to other primates.
  • Psychometric Tracery.  By tracing small fine lines on a surface, the Psychometric Tracer can leave impressions of emotions, history, knowledge, even the structure of other spells.  This is a higher level than Vibratory Telepathy as a Psychometric Tracer can share the thoughts of long-dead Tracers who lived thousands of years earlier.  By reading one Tracery and inscribing another simultaneously, Tracers can duplicate Tracings; and these replicated Tracings can even contain the detailed pattern of other spells and magics.  Thus, the Tracers wield almost unimaginable power as magicians; but Tracers can get in trouble trying to use complicated Traceries that they could not have Traced themselves.

I thought I would give a few more. First, I hereby rename The Eye (as that post’s author names this ability) to Force Perception, and I dub a user of any of these magics a Mage.

 2013-07-07

Cambridge vocab - a guide for the mystified

There is an awfully large collection of confusing words you will encounter on first coming to study at Cambridge. You pick them up really quickly in the natural run of things, but I thought perhaps a mini-dictionary might be helpful. The list is alphabetised (if I’m competent enough, anyway) and may, like so many of my writings, grow. Apologies for my crude attempts at pronunciations for the non-obvious words, but it’s very hard to find someone who can read IPA.

 2013-07-06

Cambridge undergrad maths tips

I wrote this when I was excessively bored during exam term of my first year. It may grow as I get better at working (I’m something of a revisionist). The advice is entirely Cambridge-based; a lot of it probably applies to other places with minor alterations. Most of this comes from personal experience.

During a supervision, your supervisor will be writing all the time. As soon as you leave the supervision, mark the sheets that are particularly important in some obvious way (eg. by colouring in the corner). That way, when you’re frantically flicking through the notes at the end of the year, you’ll see where the information you need is. By “most important”, I mean the places where the supervisor explains something fundamental to many questions, rather than the ins and outs of one particular question.

 2013-07-04

In which I augment the lexicon

A few dubiously-real words which I think should be more widely used.

 2013-07-03

CUCaTS Puzzlehunt

At the end of last (that is, Lent 2012-2013) term at Cambridge, I took part in the Cambridge University Computing and Technology Society Puzzlehunt (for some reason, as of this writing, they haven’t yet updated that page for this year’s Puzzlehunt, but last year’s is up there). A short summary: the Puzzlehunt is a treasure hunt around Cambridge, crossed with a whole bunch of online computing-based puzzles. It’s very difficult, and it lasts for twenty-four hours.

 2013-06-26

First post

Hello all!

In the spirit of shouting into an echoing void, this is my first post, testing whether the setup works. Some content will probably turn up soon.

 2013-06-26

Sylow theorems

A fairly long and winding way through a proof of the three Sylow theorems.

 2013-06-26
profile picture

Patrick Stevens

© Patrick Stevens 2025