# A Concept Which Has Been 'Specialized' In the Course of History

There are so many concepts which have been generalized during history of mathematics - the concept of "number" may be the best examples.

On the other hand, a concept may have been specialized ; the concept has referred to a wider range of examples in the mathematics of the past than present.

Do you know any such concept?

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It used to be not unheard of to consider "primes" to include $1$. – Henning Makholm Mar 25 '14 at 16:02
Roughly around the 1910s and 1920s the term function became specialized to what used to be known as a single-valued function. – Dave L. Renfro Mar 25 '14 at 16:28
@DaveL.Renfro Would you give your reference? – Behzad Mar 25 '14 at 16:50
The word set now has a technical meaning, out of necessity as shown by Russell. – Zhen Lin Mar 25 '14 at 17:29
@Behzad: I don't have a specific reference. Rather, this is something I've observed over many years of looking at thousands of journal articles and books from the early 1800s until the present. If it's very important to have a specific reference (for something you're working on?), I could probably find some references in a couple of days in some of the survey papers I have on the evolution of the function idea. Oh, I just noticed that I've previously posted a list of some of these papers. – Dave L. Renfro Mar 25 '14 at 18:16

As the OP correctly points out, the concept of number has been systematically generalized throughout the history of mathematics, successively extending it to the rationals, the irrationals, the imaginary numbers, and infinitesimals. Infinitesimals were uppermost in the mind of Leibniz, Euler, Cauchy, and others until the middle of the 19th century; see for example this recent study.

However, starting about 1870, and through the efforts of the great mathematicians Cantor, Dedekind, Weierstrass, and others, the concept of number was specialized to that of a real number (as well as complex, of course), to the exclusion of infinitesimals. Cantor went as far, in fact, as calling infinitesimals "the cholera bacillus of mathematics", "paper numbers", and even "an abomination". See the seminal study by P. Erhlich.

The trend was eventually reversed in the 1960s, but during the period 1870-1960 the number concept furnishes a good example of a specialisation of the meaning of a concept in the history of mathematics.

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