Impossible Problems - Arising from Conflicting Information

Impossible Problems - Arising from Conflicting Information
by Don Allen

In this second part of our series on Impossible Problems, (see http://used-ideas.blogspot.com/2013/09/impossible-problems-arising-from.html) we take up those arising from conflicts and conflicting information, and in some cases too much information.  Nonetheless, people and institutions must make decisions, regardless of the circumstances.  Doing nothing leads to conflicts and problems of another sort.

How should one deal with conflicts and/or conflicting information?  Suppose the birth date of some historical figure is in question.  There are solutions offered.  What normally is done is further research followed by a decision on which date to accept.  That done, is the issue resolved?  Indeed not, it well can be that the decision made is incorrect.  In fact, when restricted to the Internet for information sources, consider the paper by Yin, Han, and Yu in which are developed consensus ideas.  There are several possibilities for conflicting information and situations.  Here are a few.

• ·         Conflicting interpretations of the same evidence.
• ·         Conflicting but small deviations of knowledge.  Generating wide variance of solution sets.  Chaos and instability.
• ·         Conflicting information from multiple sources/conclusions. (Confusion theory? Yes.  See Camp
• ·         Primary vs. secondary resources.  Which are primary and which are secondary only begin to address this issue.

Examples include those genealogical dates, economic information and theory, scientific theories, climate change, information and disinformation, scheduling order, big data, and battle strategies.

Problems of conflicting information, particularly information of equally credible sources, often render unto us impossible problems.  

First let’s review what an impossible problem is. We’ll say a problem is impossible if its difficulty or solution cannot be imagined within the toolkit of current expertise.  A problem is called impossible if there is no possible, no conceivable, or no realizable plan to solve it.  Sure, for some problems a solution can be proposed.  Imagine a number of millions of digits.  Determine if the number is prime (i.e. has no divisors other than itself or one).  Why impossible?  This is because the number of primes less than the given number is so vast as to be virtually and permanently unknown.  Therefore, the proposed solution is unrealistic and hence the problem is classed as impossible.   The impossibility of a problem is a status; it can change.  Many impossible problems of the past are no longer impossible.  In almost every case, new paradigms of thinking were required.  

Math problems are the easiest examples to explain in a historical context.  Let’s looks at a few.  The first is the problem of squaring the circle.  This means to find using only a straight edge and compass a square having the same area as a circle of the same area.  This impossible problem took two millennia to solve – and only then in the negative.  Another was to determine the solution of a cubic equation.  This problem, this impossible problem, took a mere eighteen hundred years to solve, this time in the positive, and a bi-product was the emergence of complex numbers.    On the other hand, determining a cure for tetanus took about the same time, but required an acquired knowledge of human physiology, the germ, and the concept of vaccine to effect.   What was thought impossible, and acknowledged to be so, became possible.

Impossible or possible, we need a definition of a solution.  Normally, we should suggest what a solution to a problem is before classifying what an impossible problem might be.  This is a more-or-less an inverse approach, relying upon your informal intuition of impossibility before venturing the possible. 

Nominally, a solution suggests a resolution of a problem, which is satisfactory to general consensus.   Solving a simple math problem such as 2x + 1 = 3 leaves little room for ambiguity.  These toy problems are the exception to the rule.  We need a more robust definition, namely that of a solution set.

Definition.  A solution set is a collection of solutions to a single problem.  They may be conflicting, logically acceptable,  or similarly acceptable.  If I ask for the square root of nine, one person may offer up three, and another may offer up minus three.  Both are correct, thus making the solution set ±3.  This is the solution set for the simple problem given. 

However, when asked to vote the subject may select Candidate A or Candidate B, making A or B the solution set.  Other examples are more complicated.  In the presence of conflicting information about the candidates, a person may or make either decision, even if one source of information is less credible than another.   Indeed, data shows that the subject may not clearly be able to discriminate the good information from the bad.

Climate change, whether it exists or not, whether it is anthropogenic or not, offers numerous sources of conflicting information, and correspondent and conflicting solutions.

How does the brain make decisions about problems, with or without conflicting information?  Sure, we all try to use logic.  In complex problems and even worse impossible problems, logic is not necessarily a useful or expedient tool.  The leaves us with instinct, intuition, faith, belief, and experience.  (See, http://used-ideas.blogspot.com/2013/02/problem-solving-your-marvelous-brain.html)  All are factors, in more-or-less equal measure used for impossible problems requiring an action.   This short piece is neither original nor final.  Many people have witnessed impossible problems for impossibly many ages.  Some have names, such as Gresham’s Law, the Stoop effect, and more. 

Conflicting information can render some problems impossible.  In the absence of definitive information, it becomes impossible to solve some problems.  Instead, these problems are often resolved by choosing between alternatives. Conflicting information does not mean missing information, making problem solving really impossible, in the sense of being not solvable.   Conflicting information is what happens when information comes in two or more forms which oppose one another as to their implication.  If Sally is told by Fred that the glass bowl is cooler that the table it sits upon while Bill tells her it only feels cooler than the table because of the relative specific heats, she has received conflicting information.   Two explanations are given for the same thing.  Alternatively, Sally touches the glass bowl and it feels cooler than the table, she receives conflicting information as her “common sense” says they should be at the same temperature.   If study data shows that Mike has completed all the homework and viewed all the videos and gets a D while Mary has scarcely viewed any videos and did no homework gets an A, this is conflicting information at least from the cause and effect viewpoint. So, there can be conflicts of 

1.      Differing accounts or explanations of a single phenomenon.

2.      Observing phenomena that conflict with common sense, or one’s sense of common sense.  

3.      Logical conclusions not matching with or derivable from extant information.

Let’s consider the third example in some detail as the conflicting information involves situations where logical conclusions cannot be accurately inferred from data.

Learning analytics.

The panacea for understanding student learning is currently a topic called learning analytics.  That is, we study how a student learns from the aspect of tracking information about how the student proceeds through the course.  Vast data are determined.  Called “big data” this is what we must call a technical approach to the theory of student learning.  It removes the teacher from the equation completely – except possibly in a comparative manner.  It is akin to the technical stock market analyst, who regards only a time series of stock data and attempts to derive conclusions and predictions.  Such folks, often call technical analysts, have the disadvantage of being sometimes correct.  However, knowing some predictive analysis will ultimately change the market functionality.  While not the case with learning data, it is as well only a single facet of the learning equation. 

To my mind, if analytics was the solution to either problem, data would lead us to the importance of having such knowledge.  Both problems, learning and market operations, are essentially impossible.   It is conceded that analytics have and will provide must valuable information about their respective disciplines.  Learning analytics is just the newest of technical solutions to highly non technical problems.  They may provide interesting answers to impossible problems.  But more effort must be focused on both.  For student learning there are serious psychological aspects currently being considered including motivation, persistence and self-efficacy – and let us add conceptual internalization.  Measurement techniques are under development.  The impossible problem of student learning actually has a chance of being promoted to the realm of the possible.  It will required the combined and concentrated efforts of many disciplines, all working together.

Yet there is Confusion Theory. Many examples are temporally based, but not all. In mathematical learning it is little understood nor appreciated!  Let’s look at a few more sources of conflict.

Gresham's Law.   This law, simply put, states that bad money chases out the good. Named after Sir Thomas Gresham (1519-1579), an English financier, it was first mentioned in 1858 by Henry Dunning Macleod.   "Good" money is money that shows little difference between its nominal value (i.e. the face value of the coin) and its commodity value.  But when the money is diluted by diminishing the percentage of precious metal in it, the money becomes "bad" and the good money is sequestered by owners for another day.  Bad money has been shown historically to create economic disasters. There is a political counterpart, to wit when a credible source suggests the welfare-improving choice and a less credible source simultaneously suggests an alternate choice that will make subjects worse off, subjects make worse decisions than when only the credible source is available.  Bad information has an effect.  When conflicting information about various political candidates is presented, it is not evident that the voter will discount the lesser source(s) of credibility.   This implies that the obvious not exactly not that obvious.  Like the dilution of the precious metal, conflicting information dilutes the truth with the consequence that some citizens will be attracted to alternative, less reasonable problem solutions.

In conflicting information situations, often the decision-maker will make an incorrect decision than when only the more credible information is offered.  This implies a negative discrimination effect.

Even with the media, conflicts create confusion and skepticism. We define Media skepticism as the degree to which individuals tend to disbelieve or discount the picture of reality presented in the mass media. It  is caused in part by the process by which individuals are confronted with experiential discrepancies and what they see day-to-day.   Resultantly, they discount the media portrayal. Results seem to support the hypothesis and suggest that media skepticism.  This is yet another example of conflict in information received and knowledge learned.

Stroop’s Effect.  What does the brain do in the presence of conflicting information.  It slows it down.  The brain, in the presence of conflicting information is itself conflicted.  This  effect became widely known after John Ridley Stroop, an American psychologist, who published a paper on it in 1935. 
The study is rather simplistic, called the Stoop test.   In this test, a researcher times how long it takes a test taker to say the name of a color printed in gray or black ink. If I show you the word “blue” printed in “orange” this conflicts the mind.  It is delayed in responding the word is actually “blue.” 

Philosophers can ruminate on a single topic for centuries or longer.  Often, lovers of complexity that they are, philosophers never achieve a clear resolution to a problem, much less a solution.  These folks have time on their side, only needing to frame and consider the topic and never being required to offer an actionable statement.  Most of us, despite circumstance, must make a call, must make a decision, however flawed it may be.  Stroop’s effect commands, in presence of conflicting information that slows or delays the mind’s ability to proceed, making invalid or flawed decisions their consequence.

Too Much Information.  This is a condition under which most of us live.  It is no wonder than many students restrict incoming information flow. I know of several people that receive their national and world news from the Jon Stewart show, a comedy relief program that posits political views from time to time, but in a non intensive manner. It is “clean,” humorous, and not threatening.  It is also believable. This is a low information program, but about enough for many.  In contrast, learning analytics may be or could deliver to us information we do not need.  It may give too much information; it could offer distractions.  An important research question for the practitioners is for themselves to learn which information has no importance.  Learning analytics may be an enlightenment or a distraction.  Which, we do not yet know.

Lessons to be Learned

·         Conflicting information may profoundly affect decisions.

·         Conflicting information may induce misconceptions

·         Conflicting information may overwhelm learning and understanding.

·         People do not necessarily believe in a single solution – as we promote in math classes.

·         Students are accustomed to confusion and conflict. But do they know how to resolve them?

·         Student conduct toward the disposition of math problems may themselves be in conflict with their reality.

·         How information is represented may be an important matter for our courses.

Xiaoxin Yin, Jiawei Han, and Philip S. Yu, Truth Discovery with Multiple Conflicting Information Providers on the Web, IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 20, NO. 6, JUNE 2008

Sekuler, Robert, and Albert Erlebacker,  The Two Illusions of the Mueller-Lyer, Confusion Theory Revisited, American Journal of Psychology, 84, 4, 1971.

Joseph L. Camp Jr Confusion: A Study in the Theory of Knowledge