What is the difference between discrete and continuous mathematics?

Question

I am studying computer science and this has me absolutely flummoxed. The definition I can find is that discrete data is countable and that continuous is uncountable.

Examples are given stating that integers are discrete, and real numbers are continuous. Rationals it states are controversial.

I understand topics that fall within the category of discrete mathematics, I don't get why they do. Especially set theory, what if my universal set is the real numbers? How can I have a continuous realm of values and consider my structure to be discrete?

Considering both integers and reals belong to an unlimited realm of possibilities, I fail to see how either one is considered countable; they are both infinite. How would you define discrete mathematics and how would someone recognize discrete or continuous mathematics when faced with it?

Note: My knowledge of mathematics is discrete and very limited so please explain this as you would to a 6 year old.

Answer 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).

Answer 2

Admittedly, the distinction is not always clear cut. Various fields in mathematics tend to overlap so much that it is difficult to say "this is algebra" vs. "this is topology" (or whatever). The subconscious distinction largely develops because of the techniques one uses and their "feel."

For instance, it is said that analysis is the art of taking limits. The flavor is often the same: can we do it in this simple, finite case? Yes? Apply limit theorems. Can we take this limit after this limit? No? Okay, we need more control over this error term.

Probability and PDE can also fall into the category of 'continuous' things, for obvious reasons: lots of derivatives, measure theory, infinite-dimensional spaces, and so on. Ironically, most things in modern analysis/PDE aren't continuous à priori.

Discrete math, on the other hand, can test you in very different ways. Combinatorics is a great example, because often you need very little background to understand the question, but the underlying techniques are very sophisticated and require experience. You might consider finite group theory discrete math, and this could be reasonable if you're considering the permutation group $S_n$, but much of analysis is done on (locally compact, Abelian, Hausdorff) groups. Sieve arguments (the kind used in the proof of the twin prime conjecture) are also very combinatorial.

You shouldn't worry too much about the distinction. As Feynman pointed out, knowing the name of something doesn't bring you closer to understanding it. Dabble, learn, and eventually you'll be able to say "this feels like a typical argument in discrete math."

I mean, how do you know the difference between fruits and vegetables? There are grey areas (bananas are berries, tomatoes are fruits), but if someone hands you something that is juicy, sweet, crisp and delicious, you can say "this is a fruit!" even though no one told you.

Answer 3

According to my dictionary, "discrete" means "constituting a separate entity or part", and I think the same usage is meant when we talk of "discrete mathematics". The best way to know what discrete mathematics is about is probably just to study some texts or papers in discrete mathematics. Some people like discrete mathematics more than continuous mathematics, and others have a mindset suited more towards continuous mathematics - people just have different taste and interests.

On the other hand, the different areas of mathematics are intimately related to each other, and the boundaries between disciplines are created artificially. You might want to read an article (available online) by Laszlo Lovasz titled "Discrete and continuous: two sides of the same?"