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Should theorems, proposition, conjectures, etc. share a common counter?

For example, is the following good, for a theorem and a proposition to share the same number?

Theorem 1. ...

Proposition 1. ...

What about sharing numbering for conjectures?

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It depends on the number of theorems and propositions, but in general I prefer shared numbering, so that it's easy to look for the statement. If each type has its own counter, will Proposition 3 be before or after Theorem 3? – egreg Sep 30 '12 at 16:40
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If you write 5 theorems and then 2 propositions and 2 more theorems and then refer to Proposition 2, it's a mess to find it. Shared counter approach doesn't suffer from this. Because it comes after from Object 1 and before Object 3 whatever the objects are. – percusse Sep 30 '12 at 16:55
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and of course, if the book is large, it is good to number them within chapter/section so that the number does not go large and you can find them more easily – tohecz Sep 30 '12 at 17:00
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A reader seeing Theorem 3 could be tempted to think there were 2 theorems before that one. But no, there was Definition 1 and then Lemma 2. Maybe you should take your favorite author/editor couple and see how they do. – gniourf_gniourf Sep 30 '12 at 17:22
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Most of the time, the distinction between Proposition, Theorem, Lemma etc. is fairly arbitrary anyway. I don't think independent numbering makes sense. In particular, all of those will go into the list of theorems, won't they? So a shared numbering will give a canonical ordering here. Definitions and Examples are a different case, for which you will also have independent lists most of the time. – Stephan Lehmke Sep 30 '12 at 22:42
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1 Answer

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To quote from a review in Mathematical Reviews:

One practical criticism applies to this book as well as a large part of contemporary mathematical production: the various statements are called by different names, such as Lemma, Theorem, Proposition, Corollary; the first three are numbered independently of each other, while the numbers assigned to corollaries are functions of several variables; in addition, numbered formulae have their own separate numeration. The strain placed on the reader by this partial ordering is obvious, but apparently readers seek vengeance on other readers when they turn into authors.

Source: I. Barsotti, MR 23#A2419

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Barsotti was one of my teachers: I still remember his great course on Projective Geometry. Of course his lecture notes had shared numbers for theorems, propositions and corollaries. – egreg Sep 30 '12 at 17:52

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