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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?

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I think it's more common to give supplementary definitions. e.g. start with "continuous function", then later realize that it really was "smooth function" that you wanted to talk about, then eventually realize that you were really thinking about "analytic function". –  Hurkyl Feb 27 '14 at 16:13
Take "number," first (positive) rationals, then (implicitly) algebraic, then refined to reals. Meanwhile added negative numbers, then also complex. And the march goes on. –  vonbrand Feb 27 '14 at 18:19
Modul was once, in the second half of the 19th century, used for what we today call ideal. Galois did not explore the roots of polynomials, but of numerical functions. Others at that time used any combination of "entire rational algebraic function", so the use of "entire" and "rational" was also refined and redefined compared with the current use. –  LutzL Feb 27 '14 at 23:15
I'd maybe formulate the question as asking for historical re-definitions of word usage. The idea that "the primes (as such) have be re-defined to not include 1" feels weird. You just defined a new mathematical entity but didn't give it a new name. Also, in a way things like sets are indirectly defined differently with each new set theory. –  NikolajK Dec 16 '14 at 16:32

4 Answers 4

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).

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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!

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On this topic, see What is the smallest prime? by Chris K. Caldwell and Yeng Xiong. –  ShreevatsaR Feb 28 '14 at 9:16
And what they called prime back then would be called irreducible today :) –  Hagen von Eitzen Dec 17 '14 at 9:11

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)
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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.

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The function $x\mapsto 1/x$ is continuous. –  Tobias Kildetoft Dec 17 '14 at 9:00
@TobiasKildetoft, Thanks, good point. –  user72694 Dec 17 '14 at 9:00

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