Mathematics, psychology and sociology, philosophy.

Friday, October 29, 2010

The contraction of curricula and galaxies

A reader asked me what I thought about the limits of defining large numbers.

Such discussions begin with specific arithmetic operations and mathematical symbols in mind, and usually focus on comparing one system (such as Conway's "chained arrow notation") to another (such as "Bowers' extended operators"). The choice of symbols and operations affects how high one can go, and such discussions usually devolve into competitive games, the limits of which are fairly well handled by the Turing machine and the Lin/Rado "busy beaver function".

But such discussions usually come out of a more universal question, which regards the limits of human thought and perception in general.

Limits of human thought and perception are apparent throughout the history of numbers and mathematics. After a survey of early human developments (such as is presented in the nearly exhaustive "Universal History of Numbers" by Georges Ifrah, ISBN 0-471-37568-3) one might notice some patterns:

  • Perception and understanding are limited by the symbols in use and the concepts they represent,
  • Mastery of a given set of concepts leads to invention of new symbols and concepts.
At any point in history, or within any specific culture, there is a specific set of ideas and symbols which creates (or perhaps reflects, or both?) a natural limit of the capacity of the mind to perceive (say) large finite numbers.

It has been the trend throughout our history that the intellectual developments of earlier generations become assimilated into the body of common knowledge and added to the standard educational curriculum. As new material is added, earlier material is often compressed and taught (usually with greater efficiency) in a shorter period. So it is that the most advanced arithmetic of the early Babylonians is surpassed by that learned by today's 8- and 9-year-old students, and most of the algebra techniques of 9th century Arabia are (typically) learned by 13- or 14-year-olds today, and so on. Both are aided by more recent developments (Indo-Arabic numerals aid arithmetic; certain new teaching methods address the abstraction of variables in algebra, etc.)

Speculating about the limits of the human mind (or brain, for reductionists) can lead to discussions that test or challenge religious beliefs. I suppose the majority opinion in most cultures would state that the human mind has some kind of ultimate limit, which can be compared to the limited physical size of the human brain. (Such a conclusion helps to distinguish believers from God, avoiding blasphemy).

A universe, assuming it is also limited in size (or a visible universe as limited by an event horizon or light cone) would therefore also have a finite limit.

The development of our culture over thousands of years is a bit like an expanding light cone. The contraction of the curriculum into ever-shorter stretches of childhood is like the Lorentz contraction of galaxies known to be much further away, and therefore seen in a remote past, when the universe and the visible universe (our view of the world and the sum total of knowledge) were both much smaller.

Friday, May 21, 2010

A random sampling of my Google queries

On the weekend of the Pac-Man doodle, a friend asked me if I had "Googled anything lately", intending simply to help me discover the playable throwback game (which at the time had sound).

Misunderstanding his request, I prepared the following snapshot of my recent search activity.

Thursday, May 20, 2010 (yesterday) 10:32 PM: "system preferences" network "dns servers"

I was finding out how to use the Google DNS servers, which were indicated on a discussion forum as a way to fix a problem with certain YouTube videos not loading.

3:49 PM: "the band" discography "pepote"

I like to have accurate date tags (for popular music, the year it was on the charts; for classical, the year of debut performance). Here I was filling some missing dates. Many albums such as greatest-hits compilations are tagged with a re-mastering or re-release date, which is meaningless for me.

Wednesday, May 19, 2010 11:26 PM: when the tide comes in all of the boats rise

A metaphor that one of the men in MDI likes to use; he had T-shirts made, and today I wondered where the quote came from. It was originally used by JFK in 1963 when he was promoting spending federal funds on the Greers Ferry Dam in Arkansas.

11:16 AM: Coercive Persuasion "foot in the door" "love bomb"

I frequently look into new ideas and concepts relating to sociology, and one area I often write about is the balance of power between the individual and the group. Here I was trying to find an old reference that I had lost.

1:04 AM: perl bigint

Discovering how to use the Math::BigInt library, which allows arbitrary-precision calculation in Perl.

Tuesday, May 18, 2010 11:53 PM: pdflatex atsutil and MacTeX

I was getting my TEX typesetting system up and running on the Mac Pro.

9:23 PM: Islands of Adventure Harry Potter

Learning about the new theme park area that is opening next month.

5:52 PM: xkcd forum playpen balls

There was an xkcd cartoon in which someone filled their room with those brightly-colored baseball-sized hollow plastic balls, I wanted to find the discussion that would reveal whether such a thing was practical (best price: roughly $8000 for a typical size bedroom).

5:20 PM: sloane integer sequences

I use this site a lot (The On-Line Encyclopedia of Integer Sequences) and this time I was too lazy to find the link in my bookmarks page, so I did a Google search instead.

2:46 PM: translate portuguese

I used Google to try to figure out how to say something in Portuguese.

2:33 PM: 19111438098711663697781258214361

This number is the first in a set of consecutive prime numbers where the difference between each one and the next is the same number (in this case, 7 primes with a difference of 210), called a "CPAP". It is one of the entries on my numbers page, and I was looking to see if it was still the record-holder for smallest CPAP-7.

Monday, May 17, 2010 5:55 PM: ffmpeg me_method dia_size

Solving a problem with the program I use to convert JPG files to MP4 video for YouTube uploads (mainly for my Gray-Scott simulations). YouTube does not like the format of the ffmpeg output (the atoms are not ordered properly for streaming) and directs users to a help/support page that is entirely irrelevant because it only addresses iMovie, Final Cut, or QuickTime.

2:59 PM: Jefferson Airplane discogs

Finding more dates of old music.

1:54 AM: ezekiel chronology and 360 days prophetic year , etc.

Filling a few details in the entry for 945000 and some related entries on my numbers page.

Wednesday, May 5, 2010

Historical Origin of "Sexism" in Archetypes

2010 May 5th

I often speak with men about archetypes and the lessons they can teach about our behaviour and group interactions. Recently, one man objected to the notion that I could claim to understand the "feminine" archetypes (such as the Maiden and Crone), while another man objected to the notion that men should be encouraged to be aware of and to embody the abilities represented by the "feminine" archetypes. I also encountered another man who objected to these other two views, and believed that all of this was the result of "sexism" in our treatment of mythology, culture, and attitudes towards all aspects of psychology and sociology.

To sort this all out, I will begin with a simple "two-sided" category system. However, I do not assign anything specifically to males or females, or to what one might call "masculine personality" or "feminine personality".

The main division I use is between "communication, perception and understanding" on the one hand, and "deduction, decision and action" on the other. Note that each occurs equally often in any living thing that exhibits behaviour, regardless of sex or gender. Also, each of these two categories includes physical, emotional and mental aspects. For example, communication can be mental (through words), emotional (facial expressions) or physical (touch, gestures, watching someone move around a room).

The first category (communication) happens between two or more people, while the other can involve a single person or more than one. If you believe in the autonomy of multiple parts of the mind (the id, ego, and super-ego, an inner child, etc.) then there is "communication" inside the mind. I consider this to be part of "decision": you are using several of your skills at the same time. Awareness of the multi-part mind is fairly recent, and is too sophisticated a concept to be relevant here.

In ancient times when story-tellers "taught" wisdom they usually did so through fables involving characters. Many of the stories that were being told concerned psychology, behaviour, ethics and morality, group interaction, and so on -- the kinds of things I am discussing when I refer to "archetypes" and why they are important.

I believe that when the story-tellers needed to discuss a lesson related to communication, they told the story with a female character. When they needed to discuss a lesson related to action, they chose a male character for their story.

What happens if a young child is given a vaugely-defined object (say an oddly-shaped piece of wood) to play with? A boy is likely to pretend the object is some sort of tool or weapon, and a girl is likely to treat it like a baby or doll. There is a big nature versus nurture debate regarding this phenomenon, but it does not need to be resolved here. The only thing we need to agree on is that this phenomenon also affected the story-tellers' choices of what characters to use in their fables. (Of course, once they made such choices, the resulting oral tradition would have helped amplify the existing gender role bias in the culture).

This use of gendered characters in fables led to a gradual accumulation of culture knowledge (some of it subconscious) linking lessons to gender-roles. These lessons covered all the areas I listed above (behaviour, morality/ethics, group dynamics, etc.).

So we end up with a body of literature (myths, fables, stories, etc.) containing lessons about behaviour, most of which can be classified into one or the other of the categories I set out above. Lessons regarding communication/perception/understanding are more likely to use female characters, and those regarding deduction/decision/action are more likely to use male characters.

The archetypes have been derived from the mythology fairly recently (e.g. by Jung, Moore and Gillette). The treatment of them as "masculine" and "feminine" is a convenience of nomenclature for those who study and understand the mythology. In general, a Jung/Moore/Gillette "masculine" archetype unifies lessons and wisdom imparted by myths/fables/stories that use male characters.

The association of these with actual male and female people (as distinguished from mythological characters) is an unfortunate accident caused by the terminology.

In other words, our current use of "male" and "female" to refer to the archetypes has no relevant connection to the use of the words "male" and "female" to refer to people -- or to the use of "male" and "female" to refer to electrical cable connectors! This is much like the treatment of such words in the east (see for example the relation between male and female in the yin and yang distinction.) It is no surprise to me that eastern thought has less trouble with the gender words.

Given the problems of "sexism" in teaching that is meant to illustrate the same psychological principles in all people regardless of sex, it might be useful to purge all gender names from the archetypes entirely -- but that will be a lot of work. Moore and Gillette describe 24 "masculine" archetypes, and there are another 24 on the "feminine" side (see my table). Nearly all of them have genderized names. That's a lot of names to change!

Monday, May 3, 2010

An "Official" Nomenclature for Large Numbers?

2010 May 3rd

A former co-worker recently told me that his son has been learning (with his help) about very large numbers, including Graham's number, and asked me "if I know of any more 'official' nomenclature [for] numbers higher than centillion".

The higher the numbers go, the less official the names get. I have written much on this in the first section of my Large Numbers page.

Most folks who ask this question want to go more than just a little bit beyond centillion (10303 or 10600). Let's use 1012345 and 101027 as examples.

The only really official nomenclature is to say, for example, "ten to the power of ten to the power of twenty-seven".

I would give the prize for "second place" to Conway and Guy, The Book of Numbers (1996) pp. 13-15, who set out the system that I describe here. Under thier system, 1012345 is "one quadrilliquattuordecicentillion" and 101027 is "ten trestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestri-gintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestriginta-trecentillitrestrigintatrecentilliduotrigintatrecentillion".

I think the Knuth -yllion system would come in third; under his system, 1012345 is "ten myllion byllion tryllion decyllion undecyllion" and 101027 is "one quinvigintyllion septemvigintyllion octovigintyllion novemvigintyllion duotrigintyllion trestrigintyllion quattuortrigintyllion quintrigintyllion quinquadragintyllion quinquagintyllion duoquinquagintyllion tresquinquagintyllion quattuorquinquagintyllion quinquinquagintyllion sesquinquagintyllion septenquinquagintyllion octoquinquagintyllion unsexagintyllion quattuorsexagintyllion quinsexagintyllion sesexagintyllion septensexagintyllion unseptuagintyllion duoseptuagintyllion treseptuagintyllion quinseptuagintyllion octoseptuagintyllion novenseptuagintyllion unoctogintyllion duooctogintyllion tresoctogintyllion sexoctogintyllion septemoctogintyllion".

As you can see, systematic names for large numbers become unwieldy if you attempt to follow the classical system of giving names to each power of 10 (or powers of 1000 like Americans do today, or of a myriad as the Greeks and Chinese did, or of a million like Chuquet).

All of the other systems I have encountered are ad-hoc, unresearched and/or poorly thought out, imitations of the Chuquet names with clumsy or inconsistent decisions regarding how to proceed once the Latin ordinal number names run out. I describe some of these here.

The names googolplex for 1010100 and googolplexplex or googolduplex for 101010100 are fairly well-known. The number 1010101010000000 appeared in a 1994 journal article by Zarko Bizaca. Going beyond these, to numbers that are unwieldy to represent even as a succession of exponents:

Several academics (mostly mathematicians like Graham) have had to invent recursive function definitions to describe large finite numerical quantities, as part of a proof of some kind. As far as I have been able to tell, each such system is incompatible with every other such system.

Jonathan Bowers seems to have given more thought to this than anyone I have read about or been in contact with. His names (like exillion, tripent, baggol, trissol, dutridecal, goppatoth, golapulus, meameamealokkapoowa, and so on) are just convenient, arbitrary nicknames for various specific examples of his array notation and its multidimensional extensions. The array notation, in turn, is shorthand for a very complex set of recursively-defined functions.

Recursively-defined functions like those Bowers develops are extremely difficult to understand, and given two different recursive definitions, it can be even more difficult to prove which produces the more quickly-growing function. I am not sure how he developed his functions but I am reasonably confident that most of his claims about them are accurate. Checking his work is well beyond my patience, if not my ability. Bowers' keen abilities of comprehension are also evident in his descriptions of multi-dimensional geometric structures ("polychora", which are like polyhedra but with more dimensions).

Tuesday, April 27, 2010

Facebook, IM (chat) and IRC Phishing

2010 April 27th

I caught a Phacebook Phisher today! It was someone impersonating a friend and asking for my phone number.
Tell-tale signs:
  • A Chat or IM or IRC msg apparently from a friend, but saying nothing specific about them or about you
  • Writing in a generic style possibly atypical of your friend's normal style
  • They reply to your messages with no delay
  • Urgent or repeated requests for info (like your phone #)
  • Logs out or goes offline after only a couple minutes delay on your part
How to reply:
  • Tell them to contact you by some other means which would require them knowing something specific (like their own email password), but
  • Don't tell them that specific thing, and don't tell them what they're asking for via the IM or chat.
Likely Hacking Method
In this particular instance I believe the hacker got a Javascript running on my friend's computer. The way this can happen to you is as follows:
  • You can visit a website which runs a Javascript (typically through a banner ad) that sticks around and later acts as a "chat relay".
  • The script waits until you are in Facebook, then opens another window that is invisible (for example, hidden below the task bar)
  • Within the hidden window, it starts a Chat with any friends who are online.
  • If a friend responds, the script (running in your browser) forwards your friend's response to the hacker (who is somewhere else on the Internet).
  • Your friend can then chat with the hacker, who impersonates you.
More details of other types of phishing are at

Wednesday, April 14, 2010

Five Dichotomies of Interpersonal Politics

2010 April 14th

First a couple limiting qualifiers:

  • I do not imagine these to be the only five, or the most important five, dichotomies. They just happen to have been on my mind lately.
  • By politics, I simply mean the ways in which people interact while deciding what to do.
These dichotomies exist as characteristics/traits of people, as ways of behaving or interacting, and as points of view -- and most are a combination of all three.

In no particular order, here they are:

1. The Personal vs Team dichotomy is strongest in situations where several people are working together and risk their individual well-being in order to achieve important results that benefit all. I discuss this and give examples in MCV03, Everything That Depends on You Depends on Your Well-Being.

2. The dichotomy between Individualism and Collectivism is similar but involves the tradeoffs between the individual(s) and a much larger entity such as an entire society. This dichotomy often factors in political ideologies, such as Ayn Rand's Objectivism (which falls on the individualist end of the spectrum) and Socialism or Communism. I have written a bit more on my Collectivism page.

3. There is a dichotomy between Task and Purpose, or between the means and the end, or between the method and the ultimate result. I think of this as a succession of several things connected in a chain. Roughly in order they are Task, Goal, Project, Mission and Purpose. They have a cause-and-effect relationship, in that each results from the ones that come before it in the chain. Also, at any point in the chain, there are typically several alternatives any of which can be used to achieve the next link in the chain. I have written a lot about this on my Priorities page.

4. That dichotomy relates closely to the dichotomy between One-Time Planning and Ongoing Re-Evaluation. This is a spectrum of individual preference; each person will tend to choose how much energy to spend on thinking and planning, and how much risk to take from the adverse effects of failing to adapt or to be flexible. I discuss this on my Decision page.

5. Finally, there is a dichotomy of management style between Hands-Off Management and Micro-Management. This appears when decisions are being made about what how to carry out a task or how to accomplish the results expected in a person's job. There is a spectrum, ranging from a completely worker-oriented extreme at which the person doing the job decides how to do it, to a completely manager-oriented extreme at which every detail is specified to the worker by another person, or perhaps by several or many others.

Sunday, March 14, 2010

It's Milktaculous

2010 March 2nd

The milk industry seems to enjoy having fun with their advertising campaigns. A brother from my college fraternity wrote:
Subject: Drink More Milk, brought to you by Canada

What's up with the crazy Vancouver dairy people?

As milk advertising goes, the above is very tame. I responded:

I see your milk stop frame animation short, and raise you a campy retro rock opera featurette.

The Battle for Milkquarious

The Battle for Milkquarious, by the California Milk Processing Board. Hosted by Creativity Online.

All-student version released after the end of the contest.

(In the contest, California high-school students submitted videos, with several prizes of grants to school arts programs.)

My frat brother friend replied:

I bow before your rock opera.     Remind me why I don't write musicals again? Oh yes, something about "copious spare time"...

And bow you should — it's nothing less than milktaculous.


I happen to be a lifetime fan of cows and all things dairy, provided that it's not too bitter or sour (why did anyone ever start making cream go sour on purpose?!?!?) and tend to think the politics of anti-dairy zealots are blown rather far out of proportion: Smoking is bad for you, and so is an excess of fat, salt, or sugar — but there are people I know who'd rather whine about the advantages of goat milk over cow milk. These aren't people with a medical issue, they just want to be anti-establishment.

Friday, March 12, 2010

What is Commitment?

2010 March 11th

I was recently asked for my opinions on the following questions:
What is a commitment? Why make them? What is important to do/be when giving your word? What is important when you are accepting another man's word? What is the best way to deal with success (kept commitments) and failure (broken commitments)?
These are common questions in a volunteer group that I participate in.
"A commitment" is simply a promise, which often engenders expectations on the part of others.
"Commitment" (in the more general sense) is the personal dimension of ownership; there is also a communal dimension of ownership which is empowerment. Total ownership cannot exist and the endeavor will fail unless both dimensions (commitment plus empowerment) are present in suitable quantities.
All commitments have a level of priority, a level of importance, and a level of urgency (see the "First Things First" part of Covey's Seven Habits of Highly Effective People). It is not always true that one of these implies either of the other two. The biggest pitfalls in dealing with commitment issues come from getting one's priorities screwed up. This applies to individuals as well as groups. In most of the interesting situations there is no simple answer to a question like "which takes priority?".
Why make commitments? Teamwork. If you are dealing with a new challenge, the communication and successful follow-through of commitments are what enable coordinated action in groups. (It is not always required for teamwork: once the team has learned how to work together on a given type of task, they can usually get it done more efficiently by dispensing with the planning, talking and micromanaging.)
What is important when making and recieving a commitment? Complete and accurate communication. Each party has to understand what the other thinks is being committed, and this has to cover as many possible future contingencies as you can find time to discuss. Consider the downsides, that's how we got 24 men to the moon without losing any of them. Cover all the bases; don't bluff someone into committing by not telling them what your expectations are for each contingency. Jobs should be cleared thoroughly and in detail, the same way companies interview employment candidates.
I have written much more on all these issues here:
And also the following shorter articles:

Monday, March 1, 2010

Vostè ha estat assimilada

2010 March 1st

Traduir al català

A friend emailed me today with a link to this image of the Eixample of Barcelona and the comment:

How do you say You have been assimilated in Catalan?

The "assimilated" quote is a reference to the fictional Borg, a notably collectivist society of virtually identical aliens that have constructed a cubelike "hive" of many roughly identical rooms and passageways.

I checked into the history of Barcelona's street layout — it turns out the identical city blocks you see in the photo are the result of deliberate planning, in 1859, by urban planner Ildefons Cerdà i Sunyer. In [1] I found:

In Cerdà's project, almost all streets were straight and distributed in a regular geometrical grid with perpendicular intersections (see Figure 1). The city blocks all had the same octagonal shape12. According to Cerdà, this regular distribution was mainly aimed at avoiding privileged building zones.

(Footnote [12] reads: There were about 1,000 blocks, each one size 113.3 m x 113.3 m.)

Figure 1 showed this 1859 map by Cerdà.

In his day Cerdà was accused of being a socialist (the goal of "avoiding privileged building zones" came from this idealism — he was trying to prevent some parts of the city being rich neighborhoods and other parts being poor), and other politicians tried to block his plan but it was adopted and mostly implemented.

In a way, socialism is a real-world expression of the Borg mentality. As perceived by the Star Trek writers, the Borg, socialism and communism all share an "everybody-work-together, everybody-benefits-equally" idealism. And in the United States, identical houses and tract developments are often seen as evidence that individuality and artistic expression have been suppressed.

So my friend's Borg comparison is quite appropriate. The title of this entry was provided by Google Translate.

Monday, February 15, 2010

Food Zealotry

2010 February 15th

My conversations with friends often end in a bizarre sequence of increasingly resolute claims followed by a sudden change of subject or silence. A typical example was today's exchange on the topic of oatmeal. It went roughly like this (I for me, F for my friend):

I: For many years I avoided oatmeal because it reminded me of my chemotherapy. I had been eating oatmeal in the mornings before the chemo and I learned to associate the subsequent nausea with the oatmeal. Oatmeal became permanently unappealing. Every 5 or 10 years I have tried it again. This last time, I found that I can eat oatmeal again!

F: That's great — oatmeal is pretty healthy. It helps you control your cholesterol levels.

I: Yeah, I heard that. And it's pretty easy to make, I use the instant oatmeal that comes in small packets.

F: Oh, no you shouldn't eat instant oatmeal.

I: What? -- Instant oatmeal is the same as non-instant oatmeal, it's just been boiled and then dried out.

(sudden silence -- a pregnant pause -- and a change in topic)

My friend couldn't go on, because he had just discovered that in fact he didn't know what he was talking about. There is one good reason to question instant oatmeal: the processing method might remove fiber content. My friend could have said that -- it's a pretty simple thing to learn and to remember. And if that were the problem, it's easy to address too -- just read the package (a commonly-cited goal is 3 or more grams of nutritional fiber per serving).

I now suspect he doesn't even know the difference between "oats" and "oatmeal" (I'll save you the embarrassment, and the trouble of looking this up: they're the same, except that oat "meal" has been crushed, cut and/or ground into smaller pieces, so you don't have to boil it as long. It is a completely mechanical process.)

In this case, the only practical upshot of our discussion was to confirm my characterization of him as a food zealot. I know lots of food zealots. Most of them are dead set on trying to get other people to do what they say: they're politicians, self-appointed leaders attempting to wield power one person at a time. The zealotry is usually driven by a sincere desire to do good and address some cultural evil; in this particular case, the cultural evil (as perceived by my friend) is a common American delusion: a simplistic belief that any food which is fast and/or easy must be unhealthy and should be avoided. (A similar delusion targets all tasty foods. The anti-tasty zealots and the anti-fast zealots must be eating a lot of grapefruit and pomegranates -- they occupy the difficult-untasty quadrant of the fruit universe).

This type of "zealotry" is part of a much larger phenomenon of political behaviour seen in inter-personal relationships. There is an abundance of negative-judgmental labels: self-made victim, drama queen, bully, manipulative, vindictive, and so on. I know a few of each.

Fortunately, one can address all of this pretty easily with a simple technique. Let's call it "The Three P's of Speech Attitude":

Personal : Speaking "personally" can be done by beginning everything you say with "I" or "my". For example, I prefer not to eat instant oatmeal. Personal statements carry one really big advantage: they are much easier for people to accept on their own merits.

Positive : This simply means replacing any negative attitudes and terminology with positives, and rephrasing questions such that the desired answer is yes. For example, I prefer to eat regular oatmeal (notice no more use of "not", and replacing the demonized "instant oatmeal" with something you actually feel good about). If I can't find a positive way to say it, I also consider saying nothing at all.

Proof : When I insist on a statement that doesn't adhere to the first two P's, then I try darn hard to be ready to prove it. Proof is in the domain of persuasive speech writing -- another P that is well beyond the scope of this little article. I read a couple articles on that topic and concluded that it would be far easier just to add the words I believe that ... to whatever I was going to say!