Edit (15 November 2017): I've just stumbled across this question, which I think is asking essentially the same thing I tried to ask below, but probably worded it more clearly - and got far more attention.

In linear algebra, any "meaningful" statement about (vectors in) a particular vector space $V$ will remain true under automorphisms of $V$, and this is the basis - no pun intended - for co-ordinate-free geometry.

More generally, any "structurally meaningful" statement about a given mathematical structure should remain true under automorphisms of the structure. Is it meaningful (or perhaps even valuable) to consider "co-ordinate-free" approaches to the study of structures outside the remit of differential geometry? I would be interested in any relevant literature.

The motivation for this question came from musing on choice functions, the identity of indiscernibles, and specifically what it means to talk about choice functions even on (hereditarily) finite sets when the elements of those sets are "structurally" indistinguishable, such as conjugate roots of a polynomial. The paper "Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and −i" by Stewart Shapiro seems like it might be relevant, but I don't have access to it.

I'm not sure if I have the ability to make this question community wiki, but it probably should be. Apologies also for the mediocre tagging; I didn't feel able to make a well-informed decision about what tags would be appropriate.

Edit: if this question isn't considered to be of a level appropriate to MathOverflow, I would be happy for it to be migrated to math.stackexchange. However, I thought there might be more people here who were able to give an informed perspective on the subject.

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    $\begingroup$ I think your sweeping opening statement is too strong. For example, positive matrices (meaning positive entries) are a well studied subject, and they are certainly "meaningful." $\endgroup$ – Christian Remling Oct 10 '16 at 16:04
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    $\begingroup$ @ChristianRemling, perhaps one could argue that, to the extent that they are studied, they are combinatorial and not really linear algebraic objects. (On the other hand, the Perron–Frobenius theorem, which is the only result about them that I know, does seem really to be about linear algebra.) Anyway, that is the magic of quotation marks; no statement involving them can be 'false'. $\endgroup$ – LSpice Oct 10 '16 at 16:23
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    $\begingroup$ One might perhaps make a distinction between "linear algebra" (in which the structure group is $GL(V)$, or in some cases subgroups such as $O(V)$, $U(V)$, or $Sp(V)$, if there is some distinguished bilinear or sesquilinear form) and "matrix algebra" (the study of the explicit ring $M_n({\bf R})$ or $M_n({\bf C})$, or of rectangular matrices). Returning to the original question: isn't category theory intended to do precisely this? $\endgroup$ – Terry Tao Oct 10 '16 at 16:26
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    $\begingroup$ @ChristianRemling A positive matrix can (and should under a base-free viewpoint) be viewed as an endomorphism that maps a certain convex cone into itself. Similarly a lot of other notions that are not basis-independent can still be formalised in a base-free way by adding additional structure (of course beyond a certain point that additional structure just becomes the choice of a basis). $\endgroup$ – Johannes Hahn Oct 10 '16 at 18:52
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    $\begingroup$ @LSpice, from what I can see the abstract is free but the full article requires a subscription. $\endgroup$ – Robin Saunders Oct 10 '16 at 19:35

Sometimes you want to know where you are.

There are lots of uses for coordinates, even in abstract structures. If nothing else, order relations are created out of using terms in the language of the structure, to determine how the structure elements and different functions on the structure are created. Also, some orders are created (as in semilattices and lattices) to provide certain functionality. A lot of my mathematical life has been devoted to upper bounds and lower bounds, so this is an excuse I proffer for countering your question.

In support of your question, there are situations where the automorphism group of a structure is rich enough, or can be enriched, to allow some coordinate free analysis to be done. For me, the most striking example is in converting Abelian groups with identity to ternary groups with one basic operation t(x,y,z) set to x - y + z, and then analyzing the structure using just that ternary operation. There are doubtless similar examples in the universal algebra literature. However, I am unsure whether you seek examples like that, or whether (as Terry Tao commented) you really want category theory.

Gerhard "Which Way To The Proofs?" Paseman, 2016.10.10.

  • $\begingroup$ The ternary operation you mention works just fine for general groups, no? $\endgroup$ – LSpice Oct 11 '16 at 1:11
  • $\begingroup$ I think inverses mess up in the nonabelian case. However, I haven't read about ternary groups in two decades. I am a poor guide to the subject. Gerhard "Definitely Not An Expert Either" Paseman, 2016.10.10. $\endgroup$ – Gerhard Paseman Oct 11 '16 at 1:33

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