Talk:Equivalent definitions of mathematical structures

Isomorphic implementations?

In the section Equivalent definitions of mathematical structures#Isomorphic implementations, the second definition of natural numbers within set theory, which uses { n } for the successor of n, is much more difficult to use than the first definition, which uses n ∪ { n }. Just think about how you would write a formula to say that n < m. Under the first definition, you can just say n ∈ m. Under the second definition what would you do? Somehow you would have to find a way to talk about a sequence of elements each of which is the singleton of the previous one. Remember that at this stage, you cannot use the natural numbers to help you since you have not yet established them and their properties. JRSpriggs (talk) 11:02, 29 July 2014 (UTC)

Yes. But anyway, this is not my invention. I just took it from Natural number#Constructions based on set theory. I did not recommend any one of these implementations for practical use. But they both have some notability in the history (and philosophy?) of mathematics, see Benacerraf's identification problem. And they both are already described in Wikipedia. Why not use them as an example? Boris Tsirelson (talk) 11:16, 29 July 2014 (UTC)

And if you bother about n < m, well, one first introduces the successor function, and then goes the well-known way (sketched in the next section "Deduced structures and cryptomorphisms" following Natural number#Properties). Boris Tsirelson (talk) 11:21, 29 July 2014 (UTC)

This article doesn't use the word "theorem" even once

This is a very interesting article, but it doesn't use the word "theorem" even once. The word "prove" is used once (in what appears to be a quote), but "proof" not at all. --50.53.34.137 (talk) 17:33, 21 October 2014 (UTC)

So what? Boris Tsirelson (talk) 17:54, 21 October 2014 (UTC)

This is a mathematics article, and theorems are the heart of mathematics. Both Pudlák and the Homotopy Type Theory book use "theorem" multiple times. Here is a quote from the latter:

"Thus, the mathematical activity of proving a theorem is identified with a special case of the mathematical activity of constructing an object—in this case, an inhabitant of a type that represents a proposition."

Do you know what that means?

BTW, you have selected some very interesting quotes for the article.

--50.53.34.137 (talk) 21:13, 21 October 2014 (UTC)

Thank you for the compliments; the words "very interesting" are now used twice on this talk page. :-)

Sure, theorems are the heart of mathematics. And nevertheless they never appear inside definitions. :-) This article is about definitions. Writing it I did not need theorems. (Though, the word "proposition" is used several times.) Do you mean any specific theorem missing? Boris Tsirelson (talk) 05:54, 22 October 2014 (UTC)

What is a structure according to Bourbaki is not clear

A procedure is described to generate sets from a base set. It consists in doing Cartesian product and power set a finite number of times. Then a few examples of elements taken from one those sets are given, but nowhere it is said what is a structure in terms of those elements.

True. But an encyclopedic article is not a textbook. It gives ideas, not all details. Is it a problem to guess the general case out of these examples? See also the end of the next subsection "Transport of structures; isomorphism". I can write the definition, but should I? Probably not. It is technical, involves several principal base sets and several auxiliary base sets. An interested reader should read the sources. Boris Tsirelson (talk) 19:04, 6 March 2018 (UTC)

It is OK. I was going to delete my comment and you answered before. I realize now that you have not defined what is a structure, but you gave examples of elements taken from a set in the scale of sets that is used to define a structure. It is abstract a bit and it is hard to see what is going on at first. I feel it could be improved, but it is well written with good examples. I should try to give specific suggestions for improvements. — Preceding unsigned comment added by 108.63.141.157 (talk) 19:11, 6 March 2018 (UTC)

Nice. After all, this article is entitled "Equivalent definitions of mathematical structures", not "Structures according to Bourbaki". Boris Tsirelson (talk) 19:15, 6 March 2018 (UTC)

One idea is to say somewhere that the scale of sets corresponds to the "space" where someone can look to pick a structure for the base sets. You first pick a set in the scale of sets, which corresponds to the signature of the structure, and then pick a specific structure with that signature. Then, given the natural bijection between the two scale of sets, the notion of isomorphism is simply that on both sides, we have picked the "same" structure with the "same" signature.

At first, if you are confuse a bit like I was, one expects some kind of bijection between the two elements that you pick, but this is because I had missed the point. It is sufficient that you have picked the same element given the induced bijection. Therefore, I feel that to be a bit more explicit about what is going on could be useful, even though, formally, you have said it with the F(U) = V. — Preceding unsigned comment added by 108.63.141.157 (talk) 19:30, 6 March 2018 (UTC)

Well... but your use of the word '"space"' in the phrase '...corresponds to the "space" where someone can look to pick a structure...' is dangerous in this context, since "space" is a set endowed with a structure...

Also, your 'signature' is rather 'echelon construction scheme'.

And please sign your messages (on talk pages) with four tildas: ~~~~. (Even better, sign in to Wikipedia.) Boris Tsirelson (talk) 20:46, 6 March 2018 (UTC)

I am providing the viewpoint of someone who start to learn about that. I guess my point was that yes, it is not obvious to get the idea from the examples and something to help the reader to extract the general point, which the examples try to convey, should be stated. I also just realized that an element within a set of the scale of sets is not used to define a structure, but a specific model. So, definitively this deserves much more explanations so that the general point is clear. Just consider the fact that the title says "Structures according to Bourbaki" and it's not clear, informally or formally, both ways, where structures fit in what is explained. I feel what is well explained is what is a scale of sets. It took me a while to see that an element within a set of a scale of sets corresponds to an instance of a structure, such as a specific group in the structure of group. It would be better if the point was said explicitly. Examples and general points go together. Dominic Mayers (talk)

Nice to meet you here. Well... but mind thethey existing terminology. As far as I understand, your "instance of a structure" is a structure, and your "structure" is probably species of structures. Otherwise the usual phrase "space is a set endowed with a structure" would be "space is a set endowed with an instance of a structure". Boris Tsirelson (talk) 06:56, 7 March 2018 (UTC)

I used a definition of structure that I read in a paper written by Brassard and Robichaud https://arxiv.org/abs/1710.01380. They say that a structure has many models, which I called instances. Based on my personal communication with them, they believe it is a very common definition of structures. Also, in many introductory texts to mathematical structures, they emphasize the difference between a structure and the specific models. This distinction is at the least necessary so that isomorphic models correspond to a unique structure. This is perhaps where the confusion lies, but I am not sure. In any case, you are right that I was not clear about the terminology, but I think there is something there that does not depend only on me. There is a difficulty which is inherent to the subject here. This is another reason why we should add some explanation above the examples, which would clarify any possible confusion of this kind. Dominic Mayers (talk) —Preceding undated comment added 21:43, 7 March 2018 (UTC)

Wow! The world is changing (while I am aged); Bell inequalities influence Bourbaki structures! Interesting turn. Boris Tsirelson (talk) 06:07, 8 March 2018 (UTC)

Actually, they say that they use the definition proposed by Gericke and Martens in 1958. These authors were not working on Bell inequalities. It's in 1964 that John Bell wrote the inequality. The work of Brassard and Robichaud is related to Bell inequalities, but makes no use of them. Dominic Mayers (talk) 14:20, 8 March 2018 (UTC)

Can we make some connection with this work: http://bergeron.math.uqam.ca/wp-content/uploads/2013/11/book.pdf ? Do we have more specific references where this is described. I would like to contribute to this page, but I need references that are publicly accessible. Dominic Mayers (talk) 17:17, 8 March 2018 (UTC)

I fail to find "Gericke and Martens 1958". [But I surely know what happened in 1964 :-) ] About Bergeron et al: they treat only structures over finite sets, and are really interested only in combinatorics... I did not see them before. All relevant sources that I happen to know are already mentioned in the article. But this matter was always my hobby; I never was an expert in it. Hardly relevant, my hobbyistic essay is available here. Boris Tsirelson (talk) 20:48, 8 March 2018 (UTC)

See also User talk:Tsirel#Structures and transportability. Boris Tsirelson (talk) 18:48, 22 March 2018 (UTC)