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I have a solid (but not great) intermediate college experience in math, but now I've started exploring higher-level mathematics, out if curiosity.

I was surprised that there's a lot more uncertainty in more advanced math topics. In both probability and graph theory, I've encountered definitions that use terms open to interpretation.

What's the explanation for this? Am I right to notice that uncertainty increases in more advanced math? Or am I misunderstanding some core concepts?

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Can you give examples of such definitions? – k.stm Jan 26 '13 at 21:55
Most likely you’re misunderstanding something. Can you give some examples? – Brian M. Scott Jan 26 '13 at 21:56
Usually it is assumed that there's at most one edge between any two vertices. In my experience with graphs, it is always explicitly mentioned when this is not the case, and so the notation is safe. In the case of multigraphs, the notation of course can no longer be used. – HSN Jan 26 '13 at 22:10
@GitGud Maybe I'm being pedantic, but the OP seems to be talking about uncertainty in mathematics (e.g. a point is an undefined term) rather than a lack of specificity in mathematical exposition (e.g. an author sometimes forgets to say a graph is simple). – Austin Mohr Jan 26 '13 at 22:18
The graph example is not an instance of uncertainty about facts; it is an instance of variation in conventions. – Michael Hardy Jan 26 '13 at 22:21
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If you look up a word in a dictionary, you will find it is defined in terms of other words. Those words, in turn, are defined in terms of yet other words, and so on, until you start finding cycles; $x$ is defined in terms of $y$, which is defined in terms of $z$, etc., etc., which is defined in terms of $x$. Despite this, we manage to communicate pretty well, by and large. In any event, our difficulties in communication are due to factors other than uncertainty in the definitions.

So it is in Mathematics. Trying to define a set is guaranteed to get you into a vicious circle sooner or later. Despite that, we all wind up with pretty much the same idea of what a set is, if we stick with it, as we stick with language. Our difficulties with Mathematics are (mostly) not due to uncertainties in the definitions.

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Across entire fields, there is sometimes disagreement or inconsistency in definitions and notation: must a "ring" have an identity? What separability and compactness properties must a "manifold" satisfy? Must it possess differential structure?

Within each individual paper or book, though, there is very little uncertainty in the definitions. (Admittedly, sometimes an author is a little less explicit than he should be about his conventions, and it takes a moment to infer them.)

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