What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred? [closed]

I often heard about this division but always in a non-formal manner. What constitutes it? Is it a limit operation? Or a fundamental distinction between countable and uncountable sets? And what math subjects can be accurately attributed to one or another type?

I really find this division very weird.

closed as not a real question by Chris Godsil, Steven Landsburg, Henry Cohn, user9072, Benjamin SteinbergOct 19 '12 at 18:01

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

• What do you find weird about it? That it appears to be possible to distinguish between the two, or that anyone takes the division seriously? – Colin Reid Oct 19 '12 at 12:25
• The first variant – Reshytko Alexander Oct 19 '12 at 16:46
• Sort-of based on a false premise, and in general a bit openended. Vote to close as not a real question. – user9072 Oct 19 '12 at 17:04

There is no real division. It is for convenience. An example in which you can test out ideas on this is with the theory of finite topological spaces. (NB every thing is finite so the answer to your last part is clear in this context.) Finite topological spaces when discrete are finite sets (full stop nothing more can be said) when $T_0$ then they begin to resemble preordered sets. There is a beautiful homotopy theory of finite topological spaces and it is (almost) as rich as ordinary homotopy theory. Check it out and the division discrete non-discrete is almost quantifiable here, yet it is also very close to not existing as partially ordered sets... that is clear that is part of discrete mathematics, whilst think of them as topological spaces and they are not discrete at all.
• You say:Everything in math should be as formal as possible ... It is not formal and, for instance, terminology is often used for intuitive reasons rather than for formal ones. That being said there are parts of mathematics that a very discrete' and other parts that are very non-discrete', but many parts mix the two, often reducing a non-discrete situation to a discrete one. Perhaps one can say (and there are definite exact meanings for this) that the continuous is the limit of the discrete. – Tim Porter Oct 19 '12 at 11:25