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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 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 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 at 23:15

2 Answers 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!

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On this topic, see What is the smallest prime? by Chris K. Caldwell and Yeng Xiong. –  ShreevatsaR Feb 28 at 9:16

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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