Archive for March, 2013

Whenever I think of numbers, I form a mental image in my head.  This is not a conscious process; it happens consistently and involuntarily.  For example, whenever I imagine the numbers 1 through 100, I see something like this:


You will note several interesting features of this mental map.  Firstly, there is always a 90° left turn at the number 20; there is always a 90° right turn at the number 100.  These two kinks are the only kinks in my mental number line; the lines are perfectly straight before zero and after 100.  Why the kinks are there is mysterious.

Notice also that the image is not to scale.  That is, 50 occurs half-way between 20 and 100 (why isn’t 60 there instead?)

Here’s another mental map I have, one that appears whenever I imagine a person’s age:


You will note that this mental image is similar to the previous one, but rotated 90° to the right.  The scale is also warped: not only in the location of 50 yrs., but in the location of 10 yrs.  I believe this stems from my childhood belief that the years from age 10 to age 20 would seem to last longer than the years from 0 to 10.

Why childhood?  Well, I’ve had such mental images for as long as I can remember; it follows that they were first “constructed” in my brain at an early age.  And there is a sort of logic to the idea that 10-20 lasts “longer” than 0-10.  After all, we don’t normally recall anything about our first 5 years or so; to a child, it’s almost as if you missed those years.  So if I am 10 years old, say, and looking back at my life so far, it won’t seem nearly as long as the decade looming in front of me.  (I must stress that I am not a neuroscientist and that this is all pure speculation.)  As for why 50 is half-way between 20 and 100, I can only conclude that I wasn’t so good at calculating averages when I was younger.  The similarity of the two mental maps is best explained by positing that one of the maps is derived from the other, although which came first I cannot say.

But still, that kink…

I only became aware very, very recently that there is a name for this phenomenon.  These maps I make are called “number forms” and they are a form of synesthesia.  I have a friend who experiences grapheme-color synesthesia, seeing letters and numbers as if they had very specific colors.  It never occurred to me that my mental number maps were a related phenomenon in any way.

Here’s how I see the months of the year:


The order is always counterclockwise.  Strangely, the months are not quite evenly distributed: July is always at the top, but December/January are level at the bottom, with the (strange) consequence that there is one more month in the first “half” of the year than the second.  I also mentally divide the year into three partitions, starting at Sept. 1, Jan. 1, and June 1.  I am confident that this partitioning is a product of having attended school (on a semester system) for 25 years of my life.

Here’s the strangest map of all, but one that has (I think) the easiest explanation:


This is how I picture the recent history of the world, from the late 1700’s to the present.  There are four kinks: at 1800, 1900, 1950, and 2000.  The three biggest wars (to an American, at least) are marked in red; 1968 is also clearly “labeled” in my mental map (obviously because it’s the year of my birth).  Again, there is a lack of scale: 1800-1900 takes up as much “space” as 1900-1950.  One might conclude that I regard the 20th century as more “important” than the 19th, since I relegate more space to the former.  But there is a simpler explanation.

I can still vividly recall a timeline of history that I saw, perhaps in the 3rd or 4th grade, that has the exact same topology as this last mental map of mine.  The years from 1800 to 1970 (or so) were graphically depicted in a timeline; there were folds at 1900 and 1950, simply to make the timeline fit on the printed page.  Above key years (such as 1939) were cartoonish drawings of world events, such as World War II or Man Lands on the Moon.  Beyond the 1970’s there was nothing.  I wish I could find this image, which I believe in some sense “triggered” this form of synesthesia; I want to say that the image was in a World Book Encyclopedia but I have no proof of this claim.

In any case, I think other forms of synesthesia may also be linked to the way we first learn certain things.  My friend (who sees colors for every letter of the alphabet) once told me the probable origin of his synesthesia.  He first learned letters and numbers through colored refrigerator magnets; the colors and letters became inextricably tied in his mind, and the connections exist to this day.  For any real neuroscientists out there, I believe this is a fruitful area for further research.

Anyway, I’d be curious to see how many other people experience “number forms”.  It doesn’t make you crazy.  After all, Sir Francis Galton called his book on the subject The Visions of Sane Persons.

But still, that kink…

Read Full Post »


I don’t like to fly.

I will do it; I have flown dozens of times before; I will do it again.  But I don’t particularly enjoy the experience.  It has nothing to do with the perceived risk of flying…I know, of course, that driving on the highway is much, much more dangerous than getting into an airplane.  No, I don’t like to fly because it seems to me a particularly stressful way to die.  Seeing the wing shear off, then having the plane spiral down, gradually…

I don’t really want to talk about flying today.  What I want to talk about is a paradox that I discovered recently.  The paradox is this: the knowledge that magic is non-existent, that the universe is indifferent, that you are not special in the grand scheme of things—that knowledge can be both a comfort and a source of fear, simultaneously.  And that’s weird to me.  Shouldn’t one’s beliefs comfort you, or not comfort you, as the case may be?

Let’s say that you do think that you’re special, that you were meant for some grand purpose.  This would certainly be a comfort if you were afraid of flying.  After all, you’d say to yourself “why should I be afraid to get into an airplane?  It won’t crash—I’m special somehow!”

Now, suppose you are superstitious, and think you are special in the opposite way: you think you’re cursed, that the world is out to get you, that God hates you, whatever.  I bet that your beliefs don’t make flying a fun experience.

But what about me?  I am a physicist, a determinist, an agnostic, a free thinker.  You know how some people describe themselves as spiritual?  Whatever the opposite of spiritual is, I am that.  There’s no guardian angel watching over my shoulder; I was not meant for some higher destiny, and the universe doesn’t care about me one way or the other.  Given such beliefs, should I be comforted when I get on a plane, or not comforted?  Therein lies the paradox.

When I get on a plane, I often start thinking, “the wing’s going to rip off in mid-flight, I just know it…that would just be my luck…you just know I’m going to be on that one-in-a-million flight that crashes into the Grand Canyon.”  But then I comfort myself by saying “there’s no such thing as ‘being unlucky’.  I am no more likely than anyone else to be on an ill-fated Grand Canyon flight.  So I should take comfort, because objectively, plane crashes are really rather rare.”

The problem is, my beliefs also make me more nervous at times.  “There is no guardian angel.  I am not remarkable in any way, so I can’t be comforted by the idea that I am somehow special.  Planes sometimes crash, and there’s certainly a chance that I will be on such a flight.”

So: should I be comforted, or not comforted?

I should be comforted if I think (in some deep recess of my brain) that there really is something called bad luck, and that I really am “cursed” in some way, because when I remember that this is nonsense, I will feel better…my belief has erased a sense of foreboding that I was irrationally experiencing.

I should not be comforted if I think (in some deep recess of my brain) that there really is something called destiny, and that I really am “blessed” in some way, and that I was meant for some higher purpose than disintegration in a smoldering pile of twisted metal, because when I remember that this is nonsense, I will feel worse…my belief has erased a sense of protection and comfort that I was irrationally experiencing.

So again: should I be comforted, or not comforted?

Well, which is greater?  (1) My irrational belief (that everyone has to some degree) that I am cursed in some way, and that there is a “doom” hanging over me… or (2) my irrational belief (that everyone has to some degree) that I am blessed in some way, and that I was meant for something more?  In the first case (the pessimistic case) I should be comforted when I come to my senses, and in the second case (the optimistic case) I should be more worried when I come to my senses.

Hence the paradox:

  1. Pessimists are comforted when they remember that the universe is indifferent;
  2. Optimists are more worried when they remember that the universe is indifferent.

On balance, I am an optimist, which means my deterministic world view gives me even less comfort than it should.

Of course, given the fact that I am an optimist, I am comforted by the fact that there’s probably something wrong with my analysis.  Only, wouldn’t that make me a pessimist?  I’m so confused…

[Image is from the movie Flight (2012), directed by Robert Zemeckis.  It’s a good movie.  You should watch it.]

Read Full Post »


He should play Erdős in a movie…

My current Erdős number is 6.  I have collaborated recently with a mathematician (Jeff Lawson) whose number is 4; thus I fully expect to be “promoted” to an Erdős number of 5 within a year.

(What’s an Erdős number?  Basically, it’s the mathematical equivalent of the “Bacon number”.  You can read about Erdős numbers here.)

But I want to blog today about a different number that (frankly) I invented yesterday.  I call it a “Google number” (GN).  Here are the rules: what is the minimum number of words, none of which are proper nouns, that you must type into Google and do a Google search, such that the first search result is about you or one of your discoveries, written by you, or is a website written/maintained by you primarily?

I am very proud to say that my Google number is 2 (more on this later).

A Google number of 1 is very, very rare.  Type “president” into Google and you get a link to President of the United States…so Obama has a Google number of 1, but this will definitely change in 2017.  Type in “basketball” and you do not get Michael Jordan.  Type in “chess” and you do not get Carlsen, Kasparov, or even Fischer.

Einstein has a Google number of 1: “relativity” gives you an article on relativity.  James Cameron also has GN=1: “avatar” sends you to a James Cameron website, although you could argue that “avatar” is too close to the proper noun “Avatar” (there’s a lot of subjectivity here).    I’d be inclined to give a 1 to Thomas Edison since “incandescent” goes to an article on the incandescent bulb.  It’s not that easy to come up with more examples.  Try it yourself.

If you think about it, even the most educated person can’t have a vocabulary of much more than 30,000 words.  So logically, there can’t really be more than 30,000 or so people in the entire world with a Google number of 1.  This is 0.0004%, or 1 in 233,333.  Pretty select company.  And of course the actual number of people with Google number 1 is much, much lower, because a lot of words are “wasted”:  try searching for “harpsichord”, for example.  The lead search result does not point to any one individual person that I can see.  (Note: for simplicity, I define “Google number” as being English-specific; if we were talking about words in Spanish we might instead define a número Google, and so forth.)

What about a Google number of 2?  In theory there could be 30,000^2 = 900,000,000 people with a 2, but in practice this is fantasy.  Remember, you can’t use proper nouns.  Getting a 2 is very tricky.  Let’s try it with a famous person, to see how hard it is.  Let’s try Stephen Hawking.  “Black hole” doesn’t work; nor does “famous physicist” or “famous astrophysicist”.  “Hawking radiation” is off the table (see: proper noun.)  Even “grand design” doesn’t work because you get a British TV show first.  “History time” is a near-miss.  Finally, success: “brief history” works.  A Brief History of Time.  Hawking has a Google number of 2.

I would contend that anyone with some internet presence will have a finite Google number.  For example, my colleague Jeff (with the Erdős number of 4) has a Google number of 3: just type in “heuristic geometric phase” and you will get a talk he gave in Toronto on our joint research.  For all I know he may have a lower Google number, but it wasn’t that hard to get 3.

Now for some shameless self-promotion: my Google number is 2.  Type in “metaphor loop”.  I’m quite proud of this for some irrational reason.

One problem with this game is that it is fluid, ever-changing.  Douglas Hofstadter could very well write a book in which “metaphor loops” were integral; soon I would find my modest paper pushed to the side.  But that will happen to Obama, too, eventually, so I’m in good company.  C’est la vie.  For now, I’ll enjoy the fact that GN(me)=2, and I’ll await the vagaries of fate.

Read Full Post »

Winston Churchill and Violet Bonham Carter now appearing in: Many Worlds Comix #4!





Read Full Post »