How to prove that you are a god

I came across an interesting question while reading the blog of Scott Aaronson today. The question was as follows:

In the world of the colour-blind, how could I prove that I could see colour?

I’m presuming, to make the discussion more life-like and less cheaty, that this civilisation hasn’t discovered that light comes in wavelengths, or that it has but it can’t distinguish very well between wavelengths (so that all coloured light falls into the same bucket of 100nm to 1000nm, for instance). The challenge is to design an experimental protocol to confirm or deny that I have access to information that the colour-blind do not. This question is much harder than the corresponding question in the world of the blind, because having vision tells you so much more than having colour vision (simply set up a flag two miles away, have someone raise it at a random time, note down the time you saw it raised, and compare notes).

 2013-09-21

Stumbled across 14th September 2013

 2013-09-13

Slightly silly Sylow pseudo-sonnets

This is a collection of poems which together prove the Sylow theorems.

Notes on pronunciation

  • Pronounce \( \vert P \vert \) as “mod P”, \(a/b\) or \(\dfrac{a}{b}\) as “a on b”, and \(=\) as “equals”.
  • \(a^b\) for positive integer \(b\) is pronounced “a to the b”.
  • \(g^{-1}\) is pronounced “gee inverse”.
  • “Sylow” is pronounced “see-lov”, for the purposes of these poems.
  • \(p\) and \(P\) and \(n_p\) are different entities, so they’re allowed to rhyme.

Monorhymic Motivation 1

Suppose we have a finite group called \(G\).
This group has size \(m\) times a power of \(p\).
We choose \(m\) to have coprimality:
the power of \(p\)’s the biggest we can see.
Then One: a subgroup of that size do we
assert exists. And Two: such subgroups be
all conjugate. And \(m\)’s nought mod \(n_p\),
while \(n_p = 1 \pmod{p}\); that’s Three.

 2013-08-31

Topology made simple

I’ve been learning some basic topology over the last couple of months, and it strikes me that there are some very confusing names for things. Here I present an approach that hopefully avoids confusing terminology.

We define a topology \(\tau\) on a set \(X\) to be a collection of sets such that: for every pair of sets \(x,y \in \tau\), we have that \(x \cap y \in \tau\); \(\phi\) the empty set and \(X\) are both in \(\tau\); for every \(x \in \tau\) we have that \(x \subset X\); and that \(\displaystyle \cup_{\alpha} x_{\alpha}\) is in \(\tau\) if all the \(x_{\alpha}\) are in \(\tau\). (That is: \(\tau\) contains the empty set and the entire set; sets in \(\tau\) are subsets of \(X\); not-necessarily-countable unions of sets in \(\tau\) are in \(\tau\); and finite intersections of sets in \(\tau\) are in \(\tau\).) We then say that \((X, \tau)\) is a topological space.

 2013-08-26

Stumbled across 24th August 2013

 2013-08-24

How to punt in Cambridge

When in Cambridge…

The river is always full of beginners and professional puntists. The beginners veer all over the place, getting very wet, while the professionals zip between them, somehow managing to avoid collision by the width of an otter’s hair. The worst attempt by a beginner I’ve ever seen at punting was an attempt to use the pole rather like an oar, without ever touching the bottom of the river with it. This patent perplexity pertaining to the point of the punt provoked a pertinent post.

 2013-08-22

My experiences with flow

I’m in the middle of reading Flow, by Mihály Csíkszentmihályi, and so far, I love it. It describes the “flow state” of consciousness, that state of “everything is irrelevant except for the task at hand” in which time flies past without your noticing, and you don’t notice hunger or thirst or people moving around you. Flow can be induced when performing a difficult task which lies within your abilities, where immediate feedback is provided. I, at least, feel characteristically exhausted after coming out of a long period of flow - but it’s a good kind of mental exhaustion, much as the tiredness after a long swim is a good kind of physical exhaustion (in contrast to tiredness-after-a-long-day-of-doing-nothing, which feels sort of lazier and unwholesome). The Wikipedia page is a good enough explanation of flow that I will not describe it further here.

 2013-08-21

Thinking styles

All the way back into primary school (ages 4 to 11 years old, in case a non-Brit is reading this), we have been told repeatedly that “people learn things in different ways”. There were two years in primary school when I had a teacher who was very into Six Thinking Hats (leading to the worst outbreak of headlice I’ve ever encountered) and mind maps. I never understood mind maps, and whenever we were told to create a mind map, I’d make mine as linear and boxy as possible, out of simple frustration with the pointless task of making a picture of something that I already had perfectly well-set-out in my mind. I quickly learnt to correlate “making a mind map” with “being slow and inefficient at thinking”. (This was back when my memory was still exceptionally good, so I wasn’t really learning much at school - having read, and therefore memorised, a good children’s encyclopaedia was enough for me - and hence relative to me, pretty much everyone else was slow and inefficient, because I’d already learnt the material.)

 2013-08-18

Stumbled across 11th August 2013

 2013-08-11

New computer setup

Editor’s note: this is a snapshot of life in 2013-08-04. My setup has changed substantially since then.

In case I ever have to get a new computer (or, indeed, in case anyone else is interested), I hereby present the (updating) list of applications and so forth that I would immediately install to get a computer up to usability.

  • Browser: Firefox with Ghostery, HTTPS Everywhere, and NoScript (and remember to turn on Do Not Track…)
  • Mail client: Thunderbird with Enigmail
  • Messaging client: Adium on Mac, and possibly Pidgin for others - I’ve never used a non-Mac chat client. Beware: as of this writing, Pidgin stores passwords in plain text, so don’t save passwords in Pidgin.
  • Encryption: GPG (Windows, Mac, Linux)
  • Text editor: Vim
  • Memory training: Anki
  • Movie viewing: VLC
  • Screen colour muter: f.lux
  • Backup software: CrashPlan - but I also keep local backups using whatever built-in automated backup utility the OS provides
  • FTP client: FileZilla, or Cyberduck on a Mac
  • Syncing: Dropbox (but I want to get rid of this, because of privacy concerns)
  • Computational software: Mathematica
  • Music: iTunes (but I want to switch this for something not-Apple, and it has no Linux version)
  • Gaming: Steam
  • RSS reader: Currently, my RSS feed is presented in-browser, at NewsBlur.

 2013-08-04

Stumbled across 4th August 2013

 2013-08-04

On to-do lists as direction in life

Getting Things Done has gathered something of a cult following [archived due to link rot] since its inception. As a way of getting things done, it’s pretty good - separate tasks out into small bits on your to-do list so that you have mental room free to consider the bigger picture. However, there’s a certain aspect of to-do lists that I’ve not really seen mentioned before, and which I find to be really helpful.

 2013-07-30

Stumbled across 29th July 2013

 2013-07-29

Metathought

I have recently discovered the game of Agricola, a board game involving using resources (family members, stone, etc) to build a thriving farm. The game is turn-based, with the possible actions each turn being severely limited. This makes the game be in large part about optimising under constraint (the foundation of any good game). However, during gameplay I also detected a certain resonance between Agricola and the game of Magic: The Gathering, beyond the usual “constrained optimisation” theme. While I was playing Agricola, there was a kind of niggle in the back of my mind, telling me that “ooh, this is like Magic”.

 2013-07-25

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
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Patrick Stevens

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