" Let's say we have $n$ distinct points... " , you see this every time you open a geometry textbook. Why not just $n$ points ? If the points are not distinct, they are not exactly $n$ points, are they ? We don't write a set as $ \{x_1,x_2,x_3\}$ where $x_1 =x_2$, or do we ? Can you give me an example of a case in geometry, where an ambiguity might arise if one does not do so ?
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asked May 5, 2014 at 19:54
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$\begingroup$ Who says we are talking about sets? $\endgroup$– Marcin ŁośCommented May 5, 2014 at 19:55
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$\begingroup$ @MarcinŁoś I am speaking very naively. A set whose elements are these points. $\endgroup$– The very fluffy PandaCommented May 5, 2014 at 19:56
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6$\begingroup$ There are people who feel like distinctness should be assumed, and then there are people who feel the opposite way. Writing it this way avoids confusing a member of either group. $\endgroup$– rschwiebCommented May 5, 2014 at 20:00
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$\begingroup$ Exactly. It could mean set, it could mean, say, sequence. It's not clear from the context. $\endgroup$– Marcin ŁośCommented May 5, 2014 at 20:02
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$\begingroup$ There are certainly theorems that don't need to assume that in a collection of N points, none of the points share coordinates. Whether you consider that really M points for some M < N and simply have a point sharing multiple names or multiple points sharing coordinates doesn't matter - you still need some way to express those proofs. In general, you don't ever want to assume more conditions than are needed for a proof. $\endgroup$– ex0du5Commented May 5, 2014 at 20:07
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3 Answers
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This is done to clearly state that the points are distinct.
Note that it is not written that we have a set of $n$ points and so without writing that the points are distinct it may be interpreted that the statement being made is of the form $$ \forall p_{1},p_{2},...,p_{n}:\text{[statement]} $$
where the $p_{i}$ can be any points, without the restriction that they are different.
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When it comes to points in relation to curves (as well as to eigenvalues and to zeros of polynomials) neither set nor sequence conveys the right idea. The concept needed is that of bag: a set with an associated multiplicity for each element. For example, consider the points where two algebraic curves of degree $n_1$ and $n_2$ intersect. Typically there are $n_1n_2$ points of intersection (including complex values). But, in special cases, some of these points coincide, where the curves are tangent or have multiple crossings; the number of coincident points indicates the order of tangency or multiple intersection. If we only consider the points as a set, we have fewer then $n_1n_2$ of them, and information about where the tangencies or multiple crossings lie is lost. If there are $n$ points, and no tangency or multiple intersection is involved, we might say "$n$ points of single multiplicity". But this is making rather a meal of it; saying "$n$ distinct points" sounds lighter.
answered May 5, 2014 at 21:40
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Because, in the following discussion, the actual number of points occurs and needs to be referred to.
For example, it might say "Let the points be $(P_i)_{i=1}^n$."
answered May 5, 2014 at 19:58