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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?

Drosophila
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    Can you give examples of such definitions? – k.stm Jan 26 '13 at 21:55
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    Most likely you’re misunderstanding something. Can you give some examples? – Brian M. Scott Jan 26 '13 at 21:56
  • What about the set of edges of a graph? Some times its definition allows diferent edges from point $a$ to $b$ to be represented by $(a,b)$. This is of course a problem which is easily solved, but I've seen it more than once. – Git Gud Jan 26 '13 at 22:01
  • @GitGud, this is not a problem. A graph can have multiple edges or not. See here http://en.wikipedia.org/wiki/Graph_%28mathematics%29 – Sigur Jan 26 '13 at 22:06
  • @Sigur How is it not a problem representing two different objects by the same symbol? – Git Gud Jan 26 '13 at 22:08
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    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
  • The example that immediately comes to mind is the law of large numbers and the definition of a set. – Drosophila Jan 26 '13 at 22:14
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    @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
  • @GitGud - The possibility of representing two or more objects/ideas by the same notational symbol is a potential syntactical issue (if it cannot easily be resolved by our natural human ability to use context to parse the intended meaning), but that's hardly a problem for mathematics overall. We could, in principle, give different names to every different object in existence. The reason we don't is simply a matter of convenience as well as the inherent difficulty of a large body of individuals agreeing on the same universal set of conventions. – Michael Joyce Jan 26 '13 at 22:19
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    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
  • By uncertainty I mean that a definition, theorem, etc. is not (or maybe cannot be?) stated explicitly in mathematical terms, and can only be understood if one is familiar with conventions in the field. For example, addition and other simple operations seem intuitive to understand without needing explanation by mathematicians. The meaning of the law of large numbers, on the other hand, needed communication with others - it didn't feel like it had inherent physical meaning. So is this part of the nature of advanced math? Or is it due to my limits in understanding things on my own? – Drosophila Jan 26 '13 at 22:26
  • I see neither what "communication with others" means, nor why the LLN is an example of whatever uncertainty means. There are prima facie accepted principles/conventions which would fit the question (as mention before, what is a point, set, ..,; zorn's lemma; ,,,), but LLN I cannot make sense of. – gnometorule Jan 26 '13 at 22:31
  • @gnometorule Yes, the definition of a set is perhaps a better example. In your comment you shaped uncertainty in more precise terms for me: something that is understood in the context of convention. – Drosophila Jan 26 '13 at 22:43
  • @Drosophilia: If this really fascinates you, I have a hunch reading some Russel (which I haven't, only short excerpts) might be good - or even some of the popular math books by Smullyan (one I read way back culminated in a simple of Wittgenstein's impossibility theorem, in the guise of a playful riddle). – gnometorule Jan 26 '13 at 22:50
  • @gnometorule Thank you, I found "The principles of mathematics" online, here's the link if anyone else is interested: http://fair-use.org/bertrand-russell/the-principles-of-mathematics/index – Drosophila Jan 26 '13 at 22:55

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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.)

user7530
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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.

Gerry Myerson
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