Living & Learning

Things that I have seriously considered doing are:

Painting an entire wall of my apartment with blackboard paint.

Painting an entire wall of my apartment with whiteboard paint.

Making my own transparent whiteboard from plexiglass.

Painting a table top with blackboard paint.

Caffeine

In the Long Run We Are All Dead

Quoting Terry Tao, albeit with small modifications,

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

Although, it might not initially be obvious this is a problem about thinking about thinking about thinking about thinking — and so on.   As before, I will proceed with diagrams.  One argument, which is inductive, proceeds in this way: if there were only one blue-eyed villager, he would not know his eyes were blue.  Everyone else would know his eyes were blue.  Because the stranger pointed out there was an individual with blue eyes, an individual our sole blue-eyed villager had never seen,  he would immediately know it was him and kill himself the next day.

The Monty Hall Problem

When I was younger, I would often play a game with myself: He knows.  I know he knows.  He knows I know he knows. I pondered these sentences, turning them over slowly in my mind, thinking that eventually, I would have a feel for them.  Unsurprisingly, for the mind of a little boy, the examples that came to mind were adversial.  He tried to out know me, as I tried to out know him.  So, I could see this as a game of countering strategies.  When I felt I had an intuition built up, I tried for more,

I know he knows I know he knows.

He knows I know he knows I know he knows!

I know he knows I know he knows I know he knows!!

He knows I know he knows I know he knows I know he knows!!!

I know he knows I know he knows I know he knows I know he knows!!!!

There was always a point at which the sentences attained true meaninglessness.  It was at that point, I gave things a rest until next time.

A Rather Odd Bug in Wolfram Alpha

One very surprising result in Wolfram Alpha comes about if you try the rather obvious sum:

It claims that the “sum does not converge”.

If you don’t believe me, see for yourself.

Neal Stephenson on Science Fiction as a Literary Genre

Need help with Calculus or know someone who does?

Quite often, someone, perhaps in their first college mathematics class, will come to me saying they are having trouble and they will will ask me how to do better.   I used to be unsure of what to say.    The way people see a subject after having done it for a long time is not the same way that people see it when they are just starting out.

More on Collaboration with Google Wave

I recently found an article in Nature by Cameron Neylon.    There seems to be some overlap with my thoughts on Google Wave as well as many points of divergence.  You can read about it:

HERE

I have been wondering to what extent these online paradigms for doing mathematics (polymath on blogs , mathoverflow, wikipedia, the tricki and  forums) are a function of the community response.  Is polymath on WordPress exciting because of WordPress or because of having the right group of people at the right time?  I could say both are factors but that feels like avoiding the question.

The Complete Guide to Google Wave

If you are interested in collaborating on Google Wave, “The Complete Guide to Google Wave” might be a good resource for getting started. You can find it here.

An Attempt at Mapping Mathematics

Click for full version

On October 25th, Terry Tao wrote about the idea of creating a display visualizing the applications of mathematics.  Soon after, there was a lot of activity on MathOverflow, collecting possible uses of different fields of mathematics.  The original idea came from a visualization which illustrated the uses of the elements of the Periodic table.

I started writing this blog because of an overlap which I perceived between Tufte’s view of information display and mathematical thought. Tufte’s insight was mainly that effective information display requires an exacting form of clarity of expression where what is extraneous is identified as deleterious, and is therefore stripped away. In this view there is little room for ornamentation for the sake of ornamentation. Whether in computer code or in good writing, mathematical or otherwise, there is always something to be said for concise and clear expression.