I am interested in knowing a little bit more about the history of the concept of module. As far as I know, there are two primary meanings of the word in mathematics, namely, modules as derived from the rest classes in Gauss's Dissertationes and modules as defined in the analysis of complex numbers, where they stand for the length from the origin O to the point (x,y) which represents a given complex number in the Argand plane, which implies their being an absolute value. So basically, we have two ideas which are born in number theory (I am not sure how directly they relate to each other; clarification here would be welcome). In modern days, another use of the term module (and modular) has developed (with applications in many sciences and even in everyday life) which comes, it appears, from uses in computers and engineering, meaning, basically, components of a whole which are (partially) independent form each other in their setup, operating and so on...... I would like to know in which way this modern use depends on the original concepts.

Maybe the following excerpt from Ray Jackendoff's 2002 Book "Foundations of Language" (p. 221) helps you to provide me with feedback:

“People often seem to think of a modular capacity as entirely independent of the rest of the f-mind. This is part of what underlies the widespread conception of a modular language capacity as an isolated “grammar box” (Chapter 4). Domain specificity and informational encapsulation indeed seem to imply such a position. However, notice that an entirely domain-specific and informationally encapsulated module would be functionally disconnected from the rest of the (f-)mind, and could therefore serve no purpose in the larger goal of helping the organism perceive and behave. So there is a problem with such a caricature of modularity: how do informationally encapsulated modules “talk to each other”? Structure-constrained modularity provides an answer: levels of structure communicate with each other through interface modules.”

Thanks in advance.

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    $\begingroup$ The main meaning of module is a generalization of vector spaces: en.wikipedia.org/wiki/Module_(mathematics). For the history of that, see for instance A History of Abstract Algebra. $\endgroup$ – lhf Dec 10 '15 at 11:59
  • $\begingroup$ @lhf Well, that concept is due to Dedekind and then Noether worked on that....but from a chronological perspective the two concepts I mentioned are primary. If what you mean is that the use in engineering and computers derived from the meaning you ponted out, I would welcome an explanation about the relation..... $\endgroup$ – Javier Arias Dec 10 '15 at 12:19
  • $\begingroup$ You seem to be mixing up some words (though this may be due to the translation to English). A module is an object. The modulus (not the module btw) is something associated to a complex number. The term "modular" on the other hand is unrelated to either of these, and is probably most commonly a property certain things can have, namely modular forms. $\endgroup$ – Tobias Kildetoft Dec 14 '15 at 10:15
  • $\begingroup$ @TobiasKildetoft Well, assuming you are right, could you then provide some insight into the mathematical background of modularity as a property of certain things (circuits, mind, whatever...)? $\endgroup$ – Javier Arias Dec 14 '15 at 10:25

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