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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.

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    $\begingroup$ 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? $\endgroup$
    – Colin Reid
    Oct 19, 2012 at 12:25
  • $\begingroup$ The first variant $\endgroup$
    – Reshytko Alexander
    Oct 19, 2012 at 16:46
  • $\begingroup$ Sort-of based on a false premise, and in general a bit openended. Vote to close as not a real question. $\endgroup$
    – user9072
    Oct 19, 2012 at 17:04

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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.

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  • $\begingroup$ So there is no division formally speaking? Or it exists? $\endgroup$
    – Reshytko Alexander
    Oct 19, 2012 at 10:49
  • $\begingroup$ I mean a real formal basis. I know that it's just a convenience. What I want to know is there some solid ground in this question and if it exists what is it. Everything in math should be as formal as possible if it's just a convenienc but still it's being commonly used than probably there's something behind it but I don't undrestang it enough to put in a formal form $\endgroup$
    – Reshytko Alexander
    Oct 19, 2012 at 10:54
  • $\begingroup$ How are you to formalise such a division? Even where are you to draw the line? Combinatorics is often taken a synonymous with discrete mathematics, yet there is a famous paper of algebraic topology called Combinatorial Homotopy Theory. This treats cell complexes that look to be continuous objects but the way they are built up from basic cells is combinatorial.... One a differential manifold (so definitely non-discrete) a Morse function (approximately meaning with isolated singularities) decomposes the manifold as a cell complex!!! $\endgroup$
    – Tim Porter
    Oct 19, 2012 at 11:20
  • $\begingroup$ 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. $\endgroup$
    – Tim Porter
    Oct 19, 2012 at 11:25
  • $\begingroup$ But still I feel a kind of unsatisfaction. While continuous is often a limiting case of a discretness, it can produce the whole new algorithmic framework. The example that comes to mind is a calculus limit operation which create a whole new framework of integral and differential operatons, and as a consequence we can for example compute some things in a close form while in a dicrete case we can't. $\endgroup$
    – Reshytko Alexander
    Oct 19, 2012 at 16:33
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Our new Professor, Stephen Gilmore, is about to give an inaugural lecture entitled something like "Is Informatics an indiscrete science?", the point being, I think, that informatics is normally thought of as a fully discrete subject (being about finite or at worst countable structures), but that once you want to talk about time or power you need non-discrete methods such as differential equations. Yes, my immediate reaction is that it's about countable vs uncountable sets.

However, your question presupposes that the whole of maths splits into discrete and non-discrete, and I think that doesn't really reflect how the terms (term, really: discrete maths is a thing, but I'm not convinced non-discrete maths is a thing) are (is) used. Is finite group theory discrete maths? It's not what people usually have in mind, is it?

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