I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is usually found in "structuralist contexts" (find some examples in the appendix) and can be circumscribed by slogans such as:

“[...] objects [are] determined by the network of relationships they enjoy with all the other objects of their species.”
             Barry Mazur, When is one thing equal to some other thing?


Objects are determined by their position in their network of relationships.
              my formulation

Categorically inclined mathematicians tend to refer to Yoneda's lemma, and I do estimate this a lot, but it seems to neglect some facets of the idea. For example, category theorists don't seem to be very interested in conjugate objects, only in isomorphic ones (see here).

I am especially interested in the question whether and how the very concept of (position in a) network of relationships can be "entified" generally - just like many "concepts" that determine other entities are entities by themselves.


Some examples of "structuralist contexts" from Wikipedia: structuralism (in general), structuralism in the philosophy of science, structuralism in linguistics, structuralism in biology, structuralism in psychology, structuralism in sociology and last but not least: structuralism in the philosophy of mathematics.

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    $\begingroup$ Here's one I learned of recently: the structure identity principle (Aczel's phrase) states that if two mathematical structures are isomorphic, then they are the same. It is a consequence of Voevodsky's univalence axiom in homotopy type theory. $\endgroup$ – Zhen Lin Feb 8 '12 at 20:45
  • $\begingroup$ @Zhen: thanks for the hint(s). $\endgroup$ – Hans-Peter Stricker Feb 8 '12 at 20:50
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    $\begingroup$ This is an interesting question, but I think it stretches slightly beyond the purpose of this site. $\endgroup$ – Asaf Karagila Feb 8 '12 at 23:13
  • $\begingroup$ Unfortunately I don't understand the question ... perhaps someone can explain it in his own words? $\endgroup$ – Martin Brandenburg May 28 '13 at 1:18

Regarding the development of the idea of mathematical "structure" during the '30s, and thus independently from the "mainstream" structuralism, I suggest you the book :

and the paper :


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