Technical meanings clash with everyday meanings

Recently (see note [a]) on MathOverflow, Colin Tan asked [1] “What does ‘kernel’ mean in ‘integral kernel’?”  He had noticed the different use of the word in referring to the kernels of morphisms.

I have long thought [2] that the clash between technical meanings and everyday meaning of technical terms (not just in math) causes trouble for learners.  I have recently returned to teaching (discrete math) and my feeling is reinforced — some students early in studying abstract math cannot rid themselves of thinking of a concept in terms of familiar meanings of the word.

One of the worst areas is logic, where “implies” causes well-known bafflement.   “How can ‘If P then Q’ be true if P is false??”  For a large minority of beginning college math students, it is useless to say, “Because the truth table says so!”.  I may write in large purple letters (see [3] for example) on the board and in class notes that The Definition of a Technical Math Concept Determines Everything That Is True About the Concept but it does not take.  Not nearly.

The problem seems to be worse in logic, which changes the meaning of words used in communicating math reasoning as well as those naming math concepts. But it is bad enough elsewhere in math.

Colin’s question about “kernel” is motivated by these feelings, although in this case it is the clash of two different technical meanings given to the same English word — he wondered what the original idea was that resulted in the two meanings.  (This is discussed by those who answered his question.)

Well, when I was a grad student I made a more fundamental mistake when I was faced with two meanings of the word “domain” (in fact there are at least four meanings in math).  I tried to prove that the domain of a continuous function had to be a connected open set.  It didn’t take me all that long to realize that calculus books talked about functions defined on closed intervals, so then I thought maybe it was the interior of the domain that was a, uh, domain, but I pretty soon decided the two meanings had no relation to each other.   If I am not mistaken Colin never thought the two meanings of “kernel” had a common mathematical definition.

It is not wrong to ask about the metaphor behind the use of a particular common word for a technical concept.  It is quite illuminating to get an expert in a subject to tell about metaphors and images they have about something.  Younger mathematicians know this.  Many of the questions on MathOverflow are asking just for that.  My recollection of the Bad Old Days of Abstraction and Only Abstraction (1940-1990?) is that such questions were then strongly discouraged.

Notes

[a] The recent stock market crash has been blamed [4] on the fact that computers make buy and sell decisions so rapidly that their actions cannot be communicated around the world fast enough because of the finiteness of the speed of light.  This has affected academic exposition, too.  At the time of writing, “recently” means yesterday.

References

[1] Colin Tan, “What does ‘kernel’ mean in ‘integral kernel’?

[2] Commonword names for technical concepts (previous blog).

[3] Definitions. (Abstractmath).

[4] John Baez, This weeks finds in mathematical physics, Week 297.

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8 thoughts on “Technical meanings clash with everyday meanings”

  1. “linear” as in “linear function”
    … meaning a*f(x+y) =a*f(x)+a*f(y)…
    versus “linear” as in “linear equation”
    y(x) = Mx + B. bothered me for years.

    note that this y isn’t a linear *function* unless B=0.
    but that “y” *is* the name (sloppy though
    this might be) of a (so-called) linear *equation*.

    which renders the phrase “y is linear”…
    *even as spoken by teachers of maths*…
    ambiguous. very confusing for a first-year
    grad student (in at least one case known to me;.
    of course this’ll be right around the time
    i finally hunkered down and got all
    “to heck with what everybody *says*…
    let’s see what can be done with these
    *definitions* right here…”).

    then there’s “linear” in the *popular* sense
    meaning “predictable old-school stuff not
    informed by some gosh-wow pseudo-insight”
    (among many other things… in other words,
    meaning nothing at all in particular).

  2. and as long as we’re looking at it: “exponential”
    meaning “really, really, fast”. math teachers
    have to take the blame for this i suppose…

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