Fantasy Problems and Flying Carpet Solutions

One issue in teaching about big numbers is for students to understand just how big numbers can get.  But here we include not just students, but us all.  With modern computers and storage devices, we have become accustomed to some big numbers, notably megabytes, gigabytes, and even terabytes. In this brief article, we examine just how big numbers can get from innocent sounding problems.  This is combined with new ideas about fantasy problems and flying carpet solutions.   Fantasy problems are those arising in our minds from an array of possible disconnects with reality, and flying carpet solutions are those offered with disregard of possibility.  Some sound good; but fantasy may be the best descriptor.

One particular example is given about tracking of all people, not dissimilar from information you may have about how your various devices can currently track you.  It is a good teaching exercise of how numbers can get huge, and completely unmanageable.  The tools are only simple facts about the multiplication of exponential powers.   It also taxes the largest numbers for which there are actual names, e.g. billion, trillion, quadrillion….

Fantasy Problems

Given a problem for which there is essentially no verifiable information outside of emotions or beliefs, for which the meaning is unclear, or for which the understanding is at most vague, we call this a fantasy problem.  On the one level fantasy problems come from the formation of beliefs and using a pleasing imagination to the exclusion of evidence, reality, and rationality.  On another level fantasy problems arrive owing to ignorance, impracticality, misunderstanding, misconceptions, biased or faulty judgment.   Indeed, though applied to making particular conclusions, Kunda (1990) has reported that considerable research has documented ways in which people evaluate evidence in a biased manner. 

Some problems of religion may appear to be fantasy problems, and many are eager to dismiss them as exactly so.  Yet, many of these are reflective of deep internal workings and yearnings of the mind combined with beliefs, faith, and a need to understand the universe.  

Certainly fantasy problems are mixed with wishful thinking and self-deception, even at a state level.  Surely, there are some people that believe humans can fly if only the wind was just right, the arms were flapped just so, and the person fully believed in the possibility.   Some believe that there really is a utopia, and if mankind would only follow simple precepts it can be ours for all time.  Others may ask how many angels can dance on the head of a pin.  Sometimes, the problem posed is beyond complex tending toward grandiose or massive.  Sometimes the question is itself is not only fantasy-based but vaguely stated.   (Remember some problems come in the form of statements.) When such a problem is posed, responses may exhibit a nonsensical nature. 

Examples

• 1.      Who will win the NSF Superbowl this year?
• 2.      End all war. End crime in the streets.  End bigotry.
• 3.      Define the universe. Give three examples.
• 4.      From the perennial  “C” student… This year I will get all A’s.
• 5.      We want to track the location of every citizen of the world each second.  This way determining for interconnectedness of persons of interest will be searchable.

Except for the avid fan/statistician that studies the game intensely and can make a probabilistic estimate of the winner – not the winner, most agree the first is fantasy.  If it wasn’t, our interest in these kinds of sports would disappear.  The second is more serious; many might consider one or more of them as actually possible.   The third is clearly a fantasy, as the only precise definition of the universe must be “what is there.”   To say the least, this is a vague definition, as it is not actually known what is there.  The fourth is established as possible only by very rare exceptions, and thus overall is student fantasy. 

The last of these is of some interest, as some government operative may consider it a viable option for the actual tracking of people.  The procedure is simple to understand.  First embed a transmitting chip in each person, not at all difficult, and then collect the information.   Second process the information.  Numbers are involved, however.  The main formula we use is familiar: for a > 0, and b,c real,  a^b x a^c = a^b+c .  In the first table below, we show the components of how much data will be collected and stored in one year. 

Table 1 – Tracking Data

* This is the number of square foot area locations on earth, implying that it takes about  6x10¹⁵ digits of data to set coordinates to all such areas.  Note we’ve combined traditional names with the more modern prefix-notation – using bytes. The formula used: S=4πr², for the surface area of a sphere (earth)  of radius r.

Assuming you haven’t seen too many domegemegrottebytes in your daily experience, let’s take a look at what this means in terms of more familiar big, actually huge numbers.  In Table 2, each value 10ⁿ shown represents a one followed by n zeros. For example, 10⁶ = 1,000,000.

Table 2 – Orders of Magnitude

The terabyte is perhaps the size of your backup storage drive, and it is at least 100 times the random access memory of your computer.  This number is puny compared with what research from the University of California, San Diego that reported that in 2008, Americans consumed 3.6 zettabytes of information.  Or by Mark Liberman, (2003) who calculated the storage requirements for all human speech ever spoken at 42 zettabytes if digitized as 16 kHz 16-bit audio.  By the way, the largest single storage drive is about one exabyte as of this writing.  Clearly, the tracking problem is fantastical as realized by this back of the envelop calculation, though sounding realistic at first blush.  The domegemegrottebyte is a million, million times more than a zettabyte, already the largest value commonly used to measure uncommonly large quantities. 

When fantasy meets reality... It is safe to say this level of tracking will not happen any time soon.  However, by reducing the number of people tracked, say through select cell phone numbers, the time intervals, and the locations monitored, the estimated data base is easily manageable.  Indeed, recently China announce plans to do exactly this for about 17 million cell phone users in Bejing, but according to the government only to “to ease traffic and subway congestion.”  Right!

Flying Carpet Solutions
Given a problem for which an unrealistic, impractical, unfeasible, or naïve solution is offered, we call this a flying carpet solution.  Such solutions are derived for essentially the same reasons as fantasy problems, though probably ignorance and wishful thinking dominate (Bastardi 2011). 

Examples

• Problem.  The use of steel bars in prisons is damaging to the self-esteem of the prisoners.  Solve this problem.   Solutions.  Use force fields ala Star Trek.  Use an honor system for those willing to sign a pledge not to escape.

• Problem. Fix the educational mess.  Solutions most commonly offered.  (a) Change the curriculum.  (b) Put education on a firm business model.  What is remarkable is that these solutions have been offered time and again, and incredibly they have proponents.  Unrealistically, many believe such simplistic will actually work.

• Problem. Find a completely green system of transportation within a community.  Solutions offered.  Bicycles for everyone. Everyone should use a flying carpet. Use only electric vehicles.

This example could be presented in a single class period, with students gaining how easily big numbers arise in their lives. 

References

Bastardi, A.; Uhlmann, E. L.; Ross, L. . 2011. "Wishful Thinking: Belief, Desire, and the Motivated Evaluation of Scientific Evidence." Psychological Science no. 22 (6):731–732.

Kunda, Z. 1990. "The case for motivated reasoning." Psychological Bulletin no. 108:480-498.

Lieberman, Mark. 2014. Zettascale Linguistics  20032014]. Available from http://itre.cis.upenn.edu/~myl/languagelog/archives/000087.html.