Evolution of the definition of function

By | December 14, 2009

How do you define function? Do you teach relation first before teaching function?  Does knowing about relation a pre-requisite to function understanding?

The concept of function “was born as a result of a long search after a mathematical model for physical phenomena involving variable quantities” (Sfard, 1991, p. 14). In 1755, Euler (1707-1783) elaborated on this conception of function as a dependence relation. He proposed that, “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (p. 15). Seventy-five years later, Dirichlet (1805-1859) introduced the notion of function as an arbitrary correspondence between real numbers. About a hundred years later in 1932, with the rise of abstract algebra, the Bourbaki generalised Dirichlet’s definition. Thus, function came to be defined as a correspondence between two sets (Kieran, 1992). This formal set-theoretic definition is very different from its original definition. Function is no longer associated with numbers only and the notion of dependence between two varying quantities is now only implied (Markovits, Eylon, & Bruckheimer, 1986). The Direchlet-Bourbaki definition allows function to be conceived as a mathematical object, which is the weakness of the early definition. However, the set-theoretic definition is too abstract for an initial introduction to students and is inconsistent with their experiences in the real world (Freudenthal, 1973; Leinhardt, Zaslavsky, & Stein, 1990; Sfard, 1992).

Textbooks, which often define function as a set of ordered pairs usually start the discussion with relation and introduce function as a special kind of relation. But relation is more abstract than function. Thus the supposed pedagogical value of having to learn relation first before one understands function is, in the opinion of Thorpe (1989), wrong. Freudenthal (1973) also expressed strongly that “to introduce function, relations can be dismissed” (p. 392). Thorpe went on to say that the use of the set-theoretic definition which defines function as a set of ordered pairs “was certainly one of the errors of the sixties and it is time that it were laid to rest” (p. 13). Amen to that.

My references:

Freudenthal, H. (1973). Mathematics as an educational task.  Dordrecht-Holland:  Reidel.

Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1-64.

Markovits, Z., Eylon B. A., & Bruckheimer, M. (1986). Function today and yesterday. For the Learning of Mathematics,6(2) 18-28.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). (MAA Notes no. 25) Washington DC: Mathematical Association of America.

Thorpe, J. (1989). Algebra: What should we teach and how should we teach it? In S. Wagner & C. Kieran (Eds.) Research issues in the learning and teaching of algebra (pp. 11-24). Reston, VA: NCTM.

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11 thoughts on “Evolution of the definition of function

  1. Alan Cooper

    It is interesting to see that (if the quote has been translated correctly) Euler got it “backwards” since a change of the indep var (his “latter”) does NOT entail a change of the dep var (his “former”).

    It is also interesting to note that Fourier in the early 1800s was already pretty close to the modern level of generality (for the case of real vraiables) with the following:
    “In general, the function f(x) represents a succession of values or ordinates each of
    which is arbitrary. An infinity of values being given to the abscissa x, there are an
    equal number of ordinates f(x). All have actual numerical values, either positive
    or negative or null. We do not suppose these ordinates to be subject to a common
    law; they succeed each other in any manner whatever, and each of them is given
    as if it were a single quantity.”

    This quote is included, along with many other aspects of the story, in Israel Kleiner’s comprehensive article in the College Math Journal (1989) which is online at http://www.maa.org/pubs/Calc_articles/ma001.pdf

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  4. Whit Ford

    You never mention the notion that a function maps a given input value to a single output value. Whereas any “correspondence between two sets” will be a relation, not all are functions. A relation can map an input value to multiple outputs. I perceive this as the critical distinction between functions and relations, one that is necessary to establish before the notions of continuity and differentiability can be tackled without (additional) potential confusion. I have not researched the history as you have, so I would be interested in learning when this distinction came to be accepted.

    1. erlines

      Maybe I should have. I assumed the readers already know that. What I was trying to point out in my post are the two notions of function: dependence vs correspondence and that the former is easier to make sense of for students initially hence we should not start our teaching with the notion of correspondence.

      It was not mentioned explicitly when function was formally defined function with that condition that for every x, there’s only and only one y in the materials I considered. I think right at the very start that condition is already there because there lies the power of this mathematical model. It would not be a very useful model for predicting if not for that. My theory is that function came before relation in the consciousness of mathematicians. Relation, I suppose was born during the time when they want to define function formally and logically. Definitions of concepts should always be part of a bigger set. Hence we have “function is a relation ….” Indeed it’s more mathematical sounding definition if you say that.

      1. Whit Ford

        Another potentially interesting angle to the “function vs relation” concept would be to take a statistical approach: statistics is full of correlations, but proving cause and effect is a completely different matter. Correlations without cause & effect are comparable to relations. Correlations WITH cause & effect are functions providing that any differing output values for a given input are measurement errors (i.e. correlations that are not perfect are due entirely to measurement errors).

        I guess the notion of a function producing one output for a given input seems more intuitive to me than the notion of a relation… then again, I had been programming computers for several years before functions were introduced in math class – so my thinking was already warped into seeing the world around me as a series of functions…

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  6. moi gonowon

    do you have suggestions as to how to introduce the concept of functions?

  7. Mohammad Elsheimy

    thank u prof. i’m a newbie in math. can u recommend a book for me?
    i need a good book that discusses math from the ground up.

    thank u in advance



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