## Evolution of the Mapping/Function Concept

Hello! I'm looking for a survey (of the history) of the concept of mapping/function. How the concept was evolving. Especially I'm interested in what it turned into during the last 50 years.

So actually what are the modern views on this entity and how they emerged, the motivation behind.

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Look at the articles www.maa.org/pubs/Calc_articles/ma001.pdf and www.jstor.org/stable/10.2307/41133460. – KConrad Mar 31 2012 at 2:11
This is twenty years before your specified time-frame, but here's a point I find striking: there was very little time between the arrow notation for mappings (f: X --> Y) becoming popular and the advent of category theory. See the notes at the end of Chapter I of Mac Lane's Categories for the Working Mathematician, where he implies that the arrow notation only gained currencty about 1941. (Eilenberg and Mac Lane's foundational paper on categories was published in 1945.) – Tom Leinster Apr 30 2012 at 3:28
Incidentally, in the same note, Mac Lane says that in his 1942 paper with Eilenberg, "commutative diagrams appeared in print (probably for the first time)". On the other hand, something very like a commutative diagram appears on p.54 of Bertrand Russell's 1919 book Introduction to Mathematical Philosophy. (You can see a scan on p.13 of this file: maths.gla.ac.uk/~tl/phil/cuspomms.pdf .) – Tom Leinster Apr 30 2012 at 3:32

For the early history of the function concept, and the confusions, see the delightful book

J.E. Littlewood, "A mathematician's miscellany" Methuen, London, 1953. Republished as Littlewood's Miscellany, Edited B. Bollobas, Cambridge University Press, 1986.

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For Euler a function was something that was given by a formula (that you had to find ...). Also, for sequences $a_0,a_1,a_2,\dots$ he sometimes tried to find $a_{1/2}$. This forebodes holomorphic or analytic functions. The general notion of a function as associating a value to each argument is due to Dedekind and Weierstrass. I think that they were deeply influenced by developments in physical measurements at the same time which took place in Berlin (Helmholtz etc) where long lists of values of physical measurements were produced and discussed.

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