Ramble, Part 33: Function-al notation

[stoutfellow]

It would, I think, be hard to dispute that the greatest mathematician of the eighteenth century was Leonhard Euler. (It would not be impossible. Newton lived and continued publishing until 1727, though most of his work dated to the previous century, and Gauss began his spectacular career in 1796. I would regard the raising of such points as quibbling. Yes, I'm aware of the irony.) His work touched on every area of mathematics, and he was easily the most prolific writer in the history of the subject. (The Swiss Society of Natural Science began assembling his collected works in 1909, and they are not yet done. When finished, it will consist of over a hundred large folio volumes.) For the moment, though, I want to concentrate on his contributions to mathematical notation. It was he who gave the name e to the base of natural logarithms, and i to the square root of -1. The use of the letter Σ to represent summation is his also. Perhaps most consequential, though, was his introduction of functional notation - the representation of the relation between two variables x and y by the equation y=f(x).

The most immediate consequence of this is that it shifts the focus from the variables x and y to the relation between them; the relation, given a name of its own, is now treated as a thing in itself. Just as Viète's introduction of algebraic symbolism made it convenient to talk about numbers without specifying their values, Euler's functional notation opened the same door for relations between numbers - and, given the vast range of conceivable relations, this was a hugely consequential step. Functions could be manipulated algebraically - added, multiplied, raised to powers - or analytically - differentiated, integrated, inserted into infinite sums - and each of these would prove valuable. Less directly, it became possible to consider spaces of functions - geometric realms whose "points" are functions - and to quantify the similarity between functions. (By way of example, one much-studied question is this. Suppose two polynomials have coefficients which, term by term, are close together; can anything useful be said about how close together their zeros are? The answer is yes, and the precision with which such questions can be answered is remarkable.)

Arguably more important than the above is the realization that functions can be composed; that is, if a function f is applied, and another function g is then also applied, the combined effect is that of another function, the composition of f and g. (For example, if f(x)=x2 and g(x)=x+1, then g(f(x))=x2+1. Note that f(g(x))=(x+1)2; composing functions in different orders has different effects, just as it makes a difference whether you put on your shoes before or after your socks!) It has become clear that composition is the most fundamental operation which can be performed on functions; more on that when we get to the twentieth century.

There was one major issue that eventually forced itself on mathematicians, resulting directly from this notion of "function": what kind of thing is a function? That it involves a dependency of some sort - that the value of y is determined by the value of x - became clear early on (though there was a long period during which "multivalued functions" were permitted); but what kinds of relation were permissible? Just as Descartes had opened a can of worms by extending the notion of "curve" far beyond the old Euclidean limits, so too the notion of "function" was to have important ramifications. For example, the lack of clarity concerning this idea led, in a roundabout fashion, to Georg Cantor's development of set theory. Those ramifications, though, will have to wait for another day.

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Comments

[countrycousin.livejournal.com]

Thanks for the post.

In the meantime, since you never did explain the reference to your Viète post title, I have engaged in tireless research (well, I did a Google search) - was it a reference to an old skit done by Abbott and Costello (and others)?

[stoutfellow.livejournal.com]

Erm. Yes, that's the reference. (The version I've seen was on "I Love Lucy", with Lucy, of course, as the victim.) The allusion is to the next lines: "...step by step, inch by inch..." - obviously referring to a method of successive approximation!