What is the difference between discrete and continuous mathematics?

Solution 1:

The difference between countable and uncountable sets is well formalized and there is never any doubt. These are two different "sizes of infinity". You can read this page for more information on why countable is not the same as uncountable: http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

But I think one intuition which is really helpful, and also linking this with computer science, is the fact that a countable set is a set whose elements are finitely describable. For instance each integer can be written on a piece of paper, so the set of integers is countable. This makes integers manageable by computers: since you can completely describe an integer in a finite way, you can always pass it to a computer, as a finite sequence of $0$'s and $1$'s. This is also why reals are not countable: you might need to write down all the decimals, that is an infinite sequence. This makes "continuous mathematics" not well-suited for automatic treatment by computers.

Of course this is very schematic and can be further detailed, but this intuition is very important. It is possible to formally prove: "every element of $E$ contains a finite amount of information $\implies$ $E$ is countable".

Following this intuition, rationals are countable, because a rational $r$ can be given by two integers $a,b$ with $r=a/b$. This does not prevent rationals to be an important tool of analysis, because reals can be approached by rationals arbitrarily close (we say $\mathbb Q$ is $dense$ in $\mathbb R$). But most computer algorithms dealing with "arbitrary" numbers actually deal with only rational numbers.

As for the classification of maths into "discrete" and "continuous", the frontiers are really not well-defined, and everything interacts with everything else, so it is almost impossible to give a sound definition. A big part of it is subjective. At best, you have a "flavour" in some fields that is mostly discrete (like graph theory) or continuous (analysis), but in both cases, you might need also to consider the other side in order to get a good understanding (like using probability theory in graph theory).

Solution 2:

To address two of the questions raised:

1. What is a discrete set and how is this related to discrete mathematics?

Discrete sets consist entirely of isolated points, i.e. sets for which one can find a small enough neighborhood so that only one point of the set is contained in it -- the intuition is that the points are spread apart with a minimum distance between points.

Discrete mathematics is that which is done using finite methods typically using just the integers (e.g. combinatorics, elementary number theory) or at most a finite subset of the rationals, (e.g. discrete probability theory).

2. Are the rationals a discrete set?

No, the rationals are not a discrete set even though they are countably infinite. This is because the rationals have no isolated points---you can always find a nearby rational number as close as you like. This is called the Archimedean property of the rationals, and you can see it by asking for any tiny fraction, say 1/100,000, can I find a smaller one? Sure: 1/100,000,000. So there is no neighborhood that you can put around a rational point that you could not find another rational within.

However, any finite collection of rational numbers is discrete since when you only have finitely many rationals, then there is a minimum distance between any two rationals in the set, so one can find an interval smaller than this which will guarantee only one rational is in it at any given time. This means a finite set of rationals is an isolated set, and therefore discrete.