Talk:Mathematical structure/Archive 1

No mention of Tarski

The examples are quite good, but there's no reference to the formal Tarski notion of a structure for a language as a universe together with interpretations for the relation, function, and constant symbols. I would be somewhat tempted to start a new page called "Structure (mathematical logic)" or some such, but the existing page has considerable math-logic content, so that might be excessive duplication. Suggestions welcome. --Trovatore 28 June 2005 18:48 (UTC)

Structures do not need to be sets

The article propagates a common misunderstanding of mathematics. Namely, mathematics is implicitly identified with "set theory". Structures do not need to be sets, by far. There are categories which can be viewed abstractly (as a first order theory), so they are not sets, nor even proper classes. An abundance of examples of such structures exists in the mathematical literature. It also poses the question whether "mathematical structure" is an over-qualified title. Afterall, one is hard-pressed to give examples of structures, existing in thought, which are not mathematical or can not be expressed in mathematical terms (e.g. in terms of category theory). — Preceding unsigned comment added by 77.190.97.141 (talk) 17:51, 9 February 2018 (UTC)[reply]

I think they should be separate articles. I disagree with your statement that the current article has considerable mathematical-logic content. —msh210 03:47, 8 September 2005 (UTC)[reply]

So I now see that the concept I have in mind is treated on the Model theory page, and there's a link to it on the article page. I don't think that really covers the issue, though. Every model is a structure, and every structure is a model of some theory (say, the theory of that structure). Nevertheless "model" and "structure" are not really synonymous. It's a question of emphasis: When you speak of a model, you generally have in mind some fixed theory that it's a model of; when you speak of a structure, you may not. --Trovatore 28 June 2005 18:56 (UTC)

Directions for expansion

Random thoughts:

Structures are associated with symmetry groups as per Kleinian geometry, the importance of which in modern mathematics is still vastly underexplained in WP.
Joyal's category theory approach gives a beautiful and powerful notion of combinatorial structure, sometimes called structors or combinatorial species (despite the second name, these are functors), which can and should serve as the basis for an undergraduate textbook (I have prepared extensive notes for such a book), but for now we have only the monograph
- Bergeron, F.; Labelle, G.; and Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge: Cambridge University Press. ISBN 0-521-57323-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
the Joyal cycle index is an important extension of the well known Polya cycle index, which is related to vast area of problems in enumerative combinatorics; see
- Cameron, Peter J. Permutation Groups. Cambridge: Cambridge University Press. ISBN 0-521-65378-9.
Fraīssé theory, random graph, first order models of relational structures as per Cameron's book.

---CH 02:59, 12 August 2006 (UTC)[reply]

I've got an idea, too (actually, it's more of a gripe of sorts:) why doesn't this article state its relationship with the article on structures in mathematical logic more explicitly and clearly?

RandomDSdevel (talk) 23:45, 2 February 2013 (UTC)[reply]

I do think that a reference to graphs as a mathematical structure here would be quite relevant, and not only random graphs... Dorbec (talk) 18:17, 11 February 2020 (UTC)[reply]

Why Intuitionistic type theory?

Can someone explain why this article links to that? Pcap ping 18:52, 19 August 2009 (UTC)[reply]

Where does it link to it? Firefox "find" doesn't find it. --Trovatore (talk) 19:42, 19 August 2009 (UTC)[reply]

"... or more generally a type, ..." I know that Martin-Löf posits that mathematicians always work on some typed structure, e.g. a group is typed to him, but I'm not sure we should let his programme permeate other/all Wikipedia articles... Pcap ping 02:35, 20 August 2009 (UTC)[reply]