Robert's Blog

The contraction of curricula and galaxies

Posted in numbers, philosophy by mrob27 on 2010.10.29

2010 Oct 29th

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.


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Translated Quote-ry is the most bizarre form of flattery

Posted in other languages, philosophy by mrob27 on 2009.10.12

(Or: Patafísica y el cerebro de pollo Jorge Borges)

2009 Oct 12th

In 2000, I put a geeky but otherwise seemingly innocuous joke on my personal bio page. Five years later, it kept a whole blog-forum full of Spanish-speaking websurfers puzzled, and I didn’t even know it. Their discusson led me to discover something quite insightful.

The webpage is Here is how they quoted me:

09 Jun 2005 Paradojas

Cuando rompo con alguien un hueso de pollo de la suerte el deseo que pido es que se cumpla el deseo de la otra persona. A continuación puedo hacerle la observación de que, o bien el hueso, o bien el universo, deberían dejar de existir de repente.

Robert Munafo Publicado por Alvy # 9/Jun/2005 Categorías: Frases, Citas

I do not know any Spanish. Someone translated something I wrote into Spanish in order to quote me. Bizarre, but it gets better… I’ll give you my original quote first:

When I break a wishbone, I wish that the other person get their wish, then point out why the wishbone or the Universe should have spontaneously ceased to exist.

This was intended to be a joke, and a self-contradicting logic scenario (like the philosopher who encounters a liar and a truth-teller at a crossing in the road). It turns out not to be a paradox at all (as my Hispanic readers pointed out; answer below). I used an auto-translator to render my words back into English:

When someone break a chicken bone of luck I ask is the desire to fulfill the desire of the other person. Then I can make the observation that either the bone, or the universe, should suddenly cease to exist.3

Okay… so far so good. Brace yourself…

#2 Alex M.

No lo entiendo :S

I do not understand :S

#7 Alvy

La tradición es que el hueso lo parten dos personas, ambos piensan un deseo, a la que se queda el trozo más largo con el huesecillo central se le cumple el deseo, a la otra no se le cumple.

Por tanto, si una persona desea que se cumpla el deseo de la otra y gana el juego, entonces se produce la paradoja porque el deseo de la otra NO debería cumplirse pero se va a cumplir.

The tradition is that the bone is split two people, both think a wish, which is the longest piece with the central ossicle desire is fulfilled, the other is not satisfied.

Therefore, if a person wishes to fulfill the desire of the other and wins the game, then there is the paradox because the desire for the other should not be met but will be enforced.

#15 El Pollo

Devuelvanme mi hueso, y dejare existir su universo.

Give me back my bone, and his universe ceases to exist.

#18 18

Otra paradoja está en desear quedarte con el trozo más peque~no del hueso.

Y la estupidez está en desperdiciar el deseo deseando el trozo más grande.

Another paradox is wanting to stay with the smaller piece of bone.

And the stupidity is willing desire to waste the largest piece.

#20 Gaona

Si la tesis inicial fuera que solo pueda cumplirse un unico deseo (es decir que uno gane y el otro pierda), la hemos liado!

Me pido croquetas.

If the initial thesis was that only one wish can be fulfilled (ie that one win and one loss), we have bundled!

I ask croquettes.

#23 banpiro

No,venga,tratemos ahora el tema que nos preocupa mas a todos ¿los angeles follan? por que el que tengan sexo o no es secundario,lo importante es lo que hagan con el.

No, come, let us now more concern to us all: Do angels f***? that the having sex or not is secondary, what matters is what you do with it.

Along with all the above, and several other humorous references to subjects as diverse as Futurama, the Monty Hall problem, and chicken bones falling from the vastness of the starry night, was a serious discussion that resolved the “paradox” with various logic symbols, arrows and technical jargon. But such formalities are not really necessary, as the answer is quite simply stated:

The “paradox” is based on the notion that only one person can get his wish. But there is nothing that says that must be the case! If my friend and I break the wishbone, and both of us get our wish, then the universe and the chicken-bone can both be happy, to say nothing of me and my friend.

So I learned (in 2009) that (in 2000) I had failed to see the possibility of a win-win scenario, thanks to a bunch of writers (in 2005) whom I have never met.

Addendum: After writing the above, a friend replied:

Two more triumphs for [you]! Robert, you must know Jorge Luis Borges’ brilliant work and also the notoriously disrespectful work of the pataphysics society.

In response I’ll say the following:

I heard Borges’ brain is being kept alive in Argentina, connected via various electrodes to a computer (like in the “Spock’s Brain” episode of Star Trek) and is surfing the Internet while waiting for the Nobel committee to finally give him the Prize for literature… so maybe he instigated the discussion. Come to think of it, I don’t ever remember putting that quote on my webpage.

The wishbone custom is based in metaphysics, because it involves a belief in something beyond the physical world (namely, a belief that the wish is granted by someone or something that is not explainable in the physical world)

If there were an actual paradox, then I guess it would be ‘pataphysics. It definitely has the satirical element (-: However, as the Spanish blog forum points out, there is no actual paradox, because the wishbone belief does not say that the person with the short half of the bone will not get his wish.

So I think it’s just plain old logic.


3 : An earlier version of the same auto-translation service delivered the following:

When breaking up with someone a chicken bone of luck-wish I ask is that it meets the desire for another person. Then I make the observation that either the bone or the universe, should suddenly cease to exist.

The version given above is considerably closer to my original.

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