Is there anything mathematical that can be debated?

Sure. These things are often at the axiomatic level, underlying the foundations of mathematics. One fun example is whether the continuum hypothesis (CH) is true. CH states:

There is no set whose cardinality is strictly between that of the integers and that of the real numbers.

That is to say, there is an infinity that represents the number of integers (it doesn’t matter whether you include negative integers in that). There is another infinity that represents the number of real numbers. This second infinity can be shown to be larger than the first infinity in a precise and mathematically useful way; this was done, early on and most famously, by Cantor and his diagonal argument.

What CH says is that there’s no infinity “in between” those two—that the next level up from the integers is the reals. It turns out that CH is entirely independent of ZFC (Zermelo-Frankel plus Axiom of Choice), the set-theoretical basis of mathematics. This was shown in two separate stages:

• In 1940, Kurt Gödel showed that CH was consistent with ZFC.
• In 1963, Paul Cohen showed that CH’s negation (“there exists at least one set whose cardinality is between that of the integers and that of the reals”) was also consistent with ZFC.

These two results combine to show that CH is independent of ZFC: ZFC doesn’t tell us one way or the other whether CH is (provably) true or not. That fact notwithstanding, there is considerable discussion over whether CH is (Platonically) true or not, with serious arguments being put forth for one side or the other.

These arguments have a flavor to them that is a curious mixture of the theoretical and the rhetorical. On the theoretical side, it can be shown rigorously that if CH is true (or false, alternatively), then there are various consequences; but on the rhetorical side, the truth or falsity of CH is then argued on the basis of how compelling or likely those consequences are (none of which are decidable in ZFC either, obviously).

Essentially, people look at what happens if you assume CH (or its negation), and pick the side they find more appealing. If you like a clean and austere universe of sets, you tend to favor CH; if you like a rich and variegated universe of sets, you tend to favor its negation. Incidentally, both Gödel and Cohen tended to think of CH as false.

For those who think of mathematics as the domain of the provably true and provably false, it’s an interesting phenomenon.