Have any definitions in mathematics been redefined

Question

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has dimension one, a plane dimension two and so on. Hence when we define the term dimension, it is in such a manner that it matches with our natural feeling, whether that is in the area of topology or vector spaces or inner product spaces.

Now, very often it could turn out that the definition seems to include non-intuitive cases. For example a space filling curve does not match with the natural feeling of a curve, even though it is a continuous map as required by the definition of the curve.

My question is are there any examples in which the terms involved have been redefined because it was found that the previous definitions are inadequate?

Answer 1

Throughout the history of math terms have been redefined because the concepts are useful in more general settings than the ones where they were originally conceived:

1) prime (simple definition in the integers, but the idea of a prime has to be recast to work well in other rings, where a distinction occurs between what is now called "prime" and "irreducible"),

2) algebraic number (originally defined as a certain type of complex number, but when the importance of $p$-adic fields on the same footing as $\mathbf R$ and $\mathbf C$ became clearer, the term was applied more broadly)

3) group (originally it was a group of permutations, so in the finite case the existence of inverses was not even an axiom).

4) algebraic variety (originally defined over the complex numbers as a certain subset of affine or projective space over $\mathbf C$ before being generalized by Weil, Zariski, and finally Grothendieck). The term "elliptic curve" has undergone a similar change in its definition.

5) tensor product (first for finite-dimensional real or complex vector spaces, using bases, then more generally for modules that don't necessarily have a basis)

Some controversies over definitions continue to this day, e.g., whether or not a commutative ring should contain a multiplicative identity (see http://en.wikipedia.org/wiki/Ring_%28mathematics%29#History).

Answer 2

1 used to be a prime number, but today is not (in order not to break the Unique Factorization theorem). Come to think about it, Greeks do not even consider 1 a number at all!

Answer 3

As Hurkyl says, after "function" we defined "continuous function" and then "smooth" and "analytic", but before that the notion of function was changed several times. For example, the following features of functions are all "new" in the sense that older authors did not always use them:

• functions can be specified by something other than a concrete computation
• functions can be defined on other sets than "numbers", for whatever meaning of number
• functions are uniquely defined (not multi-functions; e.g. the square root "function" was considered to have both positive and negative values)
• functions are total (e.g., the square root function was considered a function from reals to reals)
• functions can be non-surjective (i.e. they have a defined co-domain, distinct from their range)

Answer 4

The very notion of continuity mentioned in the question is a good example. In the 18th century Euler understood a function to be continuous if the function is defined by a single equation. Thus modern continuous functions defined in a piecewise fashion would not be continuous to Euler.

In the 19th century Cauchy defined a continuous function $y=f(x)$ by requiring that an infinitesimal $x$-increment $\alpha$ always produce an infinitesimal change in $y$. In other words, $f(x+\alpha)-f(x)$ is also infinitesimal. This definition appears on page 34 in his textbook Cours d'Analyse printed in 1821, and in all subsequent texts where Cauchy discussed continuity.

Half a century later this definition of continuity was found to be insufficiently precise, and replaced by the definition $\forall\epsilon>0\exists\delta>0\forall x'(|x-x'|<\delta\rightarrow|f(x)-f(x')|<\epsilon)$.

In 1961 Abraham Robinson found a way of making Cauchy's definition precise, so that we can again define continuity in an intuitive fashion a la Cauchy, with some interesting twists related to uniform continuity.