Posted by: Alexandre Borovik | February 14, 2010

## Consistent with WHAT?

Two weeks ago I started to teach my regular first year course of linear algebra and discovered a new phenomenon: unlike their predecessors, the new cohort of students no longer understand the expression “This system of equations is consistent”. “Consistent with WHAT?” — I was asked. I suspect the misunderstanding  comes from the  extremely popular American TV series “Crime Scene Investigations” (a whole TV channel runs them, in several version: CSI Las Vegas, CSI Miami, CSI NY, and even some military (navy) vesion), where the word “consistent” can be heard a dozen times in every episode in phrases like “The fructures of the scull are consistent with the victim being hit over the head with an obtuse heavy object”. This is a professional legalistic slang; for the viewer, I am afraid, as a result of hundreds unchallenged repetitions, the expression “is consistent with” now means “PROVES THAT”.

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## Responses

1. Is the problem that they don’t know the concept of “consistency” (http://en.wikipedia.org/wiki/Consistency) or that they think they know that it means “proves that”?

Would it help if you would say: “This system of equations is consistent within itself”?

• “This system of equations is consistent within itself”: yes, this is what I eventually explained to my students.

2. “This system of equations has a solution.” 😛

3. The CSI usage of “is consistent with” is not merely a legalist phrase; it is common in contexts in which one is making inductive inferences. The data do not prove the hypothesis, but if they are consistent with it then the hypothesis is supported by them. (Of course merely logical consistency with the hypothesis is not enough, else every hypothesis would be supported by any irrelevant datum.)

4. Even in the context of CSI, “agrees with”, “doesn’t rule out” seem to be better interpretations of “is consistent with” than “PROVES THAT”. The reason is that a sentence like

“The fractures of the scull are consistent with the victim being hit over the head with an obtuse heavy object”

is merely saying that “the victim being hit over the head with an obtuse heavy object” and “fractures of the skull” COULD occur in that order, not necessarily that it DID or MUST. Just that one thing doesn’t rule the other out. In this sense, the meaning remains largely in tact even in the context of systems of equations. Each equation is a statement and so we can talk about one statement not ruling another statement out i.e. we can ask

“Is this equation (resp. statement) consistent with every other equation (resp. statement) in this list?”.

The consistency of a “system” is a new idea to students, I suppose. So defining the consistency of a system/list as

‘The property of a system/list where the answer to the above question is “yes” for every equation in the system – i.e. no equation rules another out.’

might make it easier to understand linear algebra without completely disagreeing with/ letting go of their beloved CSI episodes. CSI and lin. alg. can be consistent after all.

5. When I thought of moving things away from grey areas, ‘McLachlan’ isn’t what I had in mind. Oh well…

6. Or “skull” — unless you are back to N.C.I.S.

scull:
n.
1. A long oar used at the stern of a boat and moved from side to side to propel the boat forward.
2. One of a pair of short-handled oars used by a single rower.
3. A small light racing boat for one, two, or four rowers, each using a pair of sculls.

The Naval Criminal Investigative Service (NCIS) is an elite worldwide federal law enforcement organization whose mission is to protect and serve the Navy …

But I know EXACTLY what you mean, both in my university and high school Math teaching experience.

7. This is only loosely related (in that they are examples of students not understanding the language we take for granted, but they’re from the literary field):

Teaching Chaucer’s “The Wife of Bath,” and discussing the Knight that took the young girl’s “maidenhead,” more than half of the class thinks that he cut off her head, and are stunned to learn that really, he raped her.

Also, regarding 19th century novels where characters leave their calling cards — one student wrote an entire paper on the symbolism of telephones in one certain work.

And finally, one of my students wrote a paper about the significance of candy in Voltaire’s Candide, centering her discussion on the young children playing Skittles.

8. As every one else on this thread has already pointed out, one way of thinking about it that to say a system is consistent is just to say that each element of the system is consistent with all of the others. But another (more logic-centric) way is that what the system is really consistent with is the statement that the system has at least one solution.

It seems to me that there is a nice ‘teachable moment’ here, if you have the time for it in your class. Judging by your anecdote here, most of your students have not yet had their first real logic course, and since yours is intro to lin alg, many probably never will have such a course. A (short) digression at that point in the class about the law of the excluded middle and what it means for to statements to be (logically) inconsistent may be useful .

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